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r C3: c^<: cr ore <: ^ 'CiC: y^';^ -scrcixrcf coc: -^c:^^ :$^^ ;^2 CO J^^^c^-t >C^_ <'■ C'( ■ 5j^^:^ r CI A ^^^^- CON 2^1 !C^ ii^ :-. HARVARD UNIVERSITY. L I B R A R ^^ MUSEUM OF COMPARATIVE ZOOLOGY. COMPARATIVE ZOOLOGY. 'yyxLLjatw ^ S^-MIy/L lyo. '■'A TRANSACTIONS CONNECTICUT ACADEMY ARTS AND SCIENCES. VOI.XJ]ME III NEW HAVEN: PUBLISHED BY THE ACADEMY. "^1874 to 1878. Tutlle, Morehouse A Taylor, Printers, New Haven. CONTEISTTS PAGE List of Additjons to the Library,.... .. ... .... y Art. I. — Report on the dredgixgs in the regiox of St. George's Banks, ix 1872. By S. 1. Smith and O. Harger. Plates 1 -8, 1 Art. IL — Descriptions of new and rare species of Hy- DROIDS FROM THE NeW EnGLAND COAST. By S. F. Clark. Plates 9-10, _ 58 Ar'J'. III. — On the Chondrodite from the Tilly-Foster IRON MINE, Brewster, N. Y. By E. S. Dana. Plates 11-13, 67 Art. IV. — On the Transcendental curves sin y sin iny=. a %\n X ^\w nx-\-h. By H. A. Xewton and A. W. Philips. Plates 14-37, . 97 Art. V. — On the equilibrium of heterogeneous sub- stances. By J. W. GiBBS. First Part, 108 Art. VI. — The Hydroids of the Pacific coast of the United States south of Vancouver Island, with a REPORT upon those IN THE MuSEUM OF YalE ColLEGE. By S. F. Clark. Plates 38-41, 249 Art. VII. — On the anatomy and habits of Nereis virens. By F. M. Tubnbull. Plates 42-44, 265 Art. VIII. — Median and paired fins, a contribution to THE history of VERTEBRATE LIMBS. By J. K. ThACHER. Plates 49-60, 281 Art. IX. — Early stages of Hippa talpoida, with a note ON the structure of the MANDIBLES AND MAXILLA IN HipPA AND Remipes. By S. I. Smith. Plates 45-48, 311 Art. X. — On the equilibrium of heterogeneous sub- stances (concluded). By J. W. Gibbs, 343 OFFICERS OF THE ACADEMY. President, ELIAS LOOMIS. Vice-President, HUBERT A. NEWTON. Secretary, OSCAR HARGER. Librarian, ADDISON VAN NAME. Treasurer, HENRY C. KINGSLEY. Publishing Cormn ittee, HUBERT A. NEWTON, ELIAS LOOMIS, GEORGE J. BRUSH, ADDISON E. VERRILL, CHESTER S. LYMAN, WILLIAM D. WHITNEY. Auditing Committee, HUBERT A. NEWTON, DANIEL C. EATON, HENRY T. BLAKE. ADDITIOlSrS TO THE LIBRAEY From December 1, 1873, to June 1, 1876. Anierican Association for the Advancement of Science. Proceedings. Twenty-second. twenty -third Meetings. 8". Memoirs. I. 4 . Salem, 1874-5. Alr.\ny. — Institute. Transactions. Vol. VIT, VIII. 1872-6. 8°. Proceedings. Vol. I. a-4, II. 1. 1871-4. .8°. New York State Cabinet of Natural History. Twenty-third, twenty-fifth Annual Reports. 1870-2. 8°. New York State Library. Fifty-sixth Annual Report. 1874. 8°. Boston. — American Academy of Arts and Sciences. Proceedings. Vol. VIII-X. 1868-75. 8°. Society of Natural History. Memou's. Vol. II. i. 1, ii. 4, iii. 1-5, iv. 1-4. 1871-6. 4°. Proceedings. Vol. XV. 4, XVI, XVII, XVIII. 1-2. 1873-6. 8 . Henz, N. M. The Spiders of the United States. (Occasional Papers. II). 1875. 8". Buffalo. — Society of Natural Sciences. Bulletin. Vol. I. 4, II. III. 1-2. 1874-6. 8°. CAMBRIDGE. — Museum of Com'parative Zoology. Illustrated Catalogue. No. IV-VIII. 1871-4. 4°. Bulletin. Vol. II. 3-5, III. 1-14. 1871-6. 8". Annual Report. 1870-1875. 8°. 'him'HEXPOijis.-- Minnesota Academy of Natural Sciences. Bulletin. 1874. 8°. New York. — Lyceum of Natural History. Annals. Vol. XI. 3-6. 1875. 8°. PouGHKEEPSiE. — Society of Natural Science. Proceedings. Vol. I. 1-2. 1876. 8°. St. Louis. — Academy of Science. Transactions. Vol. III. 2. 1875. 8°. Sxh^M.— Essex Institute. Bulletin. Vol. V. 6-12, VI, VII. 1-7. 1873-5. 8°. Peabody Academy of Science. Memoirs. Vol. I. 4. 1875. 4\ San Francisco. — California Academy of Sciences. Proceedings. Vol. I. 1854-7 (2d ed. 187.3), III. 2. 1873. 8°. Savannah. — Georgia Historical Society. Jones, C. C, Jr. Se.'-geant Wm. Jasper. An Address before the Georgia Historical Society, Jan. 3, 1876. 8°. Washington. — Surgeon GeneraVs Office. Annual Report of tlie Surgeon General, U. S. Army. 1874. 8°. Circular No. 8. Report on the Hygiene of the U. S. Army. 1875. 4\ United States Naval Observatory. Astronomical and Meteorological Observa- tions. 1871, 1873. 4". Worcester. — A77ierican Antiquarian Society. Proceedings. No. 62-65. 1874-5. 8°. vi Additions to the Lihrary. Amsterdam. — Koninklijke Akademie van Wetensrhappen. Yerslagen en Mededeelingen. Afdeel. Natuurkunde. Tweede Reeks. Deel VII. ISTS. 8". Jaarboek. 1872. 8°. Augsburg. — Naturhistorischer Verein. Bericht XXII, XXIII. 1873-5. 8°. Basel. — Naturforschende Geselbchaft. Bericht iiber die Verhandlungen. I-YIII. 1 835- 1849. 8°. Yerhandhmgen. Theil YI. 1-2. 1874-5. 8°. B ATA VI A. — Natuurkundige Vereeniging. Natimrkundig Tijdschrift voor Nederlandsch Indie. Deel XXII, XXIII. 1871-3. s°. Societe des Arts et des Sciences. Tijdschrift. Yol. XX. 4-fi, XXI. 1-2. 1872-4. 8°. Notulen. X. 4. XI. 1-4. 1873-4. 8°. Codicum Arabicoriim ( 'atalogus. 1873. 8°. Alphabetische Lijst van Kaarten. 1873. 8°. Belfast. — Natural History and Philosophical Society. Proceedings. Session 1872-3, 1873-4, 1874-5. 8°. Berlin. — Konigliche Akademie der Wissenschaften. Physikalische Abhandlungen. 1838, 1841, 1842. 1845, 1849. 4°. Mathematische Abhandlungen. 1845. 4°. Bericht iiber die Yerhandhmgen. 1854-5. 8°. Monatsbericht. 1856-9. 8°. Bologna. — Accademia delle Scienze delV Instituto di Bologna. Rendiconto. 1873-4. 8". Bonn. — Naturhistorischer Verein der preussischen Rheinlande und Westplialens. Yer- handlungen. Jahrg. XXIX. 2, XXX, XXXI, XXXII. 1. 1872-5. 8°. Bordeaux. — Societe des Sciences Physiques et Naturelles. Memoires. T. IX, X, II. Ser. I. 1. 1873-5. 8°. Societe Linneenne. Actes. T. XXYII. 2, XXYIII. 1872. 8°. Bremen. — Naturwissenschaftlicher Verein. Abhandlungen. Bd. III. 4, IV. 1. 1873-4, 8°. Beilage. No. 3. 1873. 4°. Brunn. — Naturforscher Verein. Yerhandhmgen. Bd. Y, VI, XI, XII. 1866-74. 8°. Brussels. — Academie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, Memoires. T. XL. 1873. 4°. Memoires Couronnes et Memoires des Savants I^trangers. T. XXXYII, XXXVIII. 1873-4. 4°. Memoires Couronnes et Autres Memoires. T. XXIII. 1873. 8°. Bulletins. II. Ser. T. XXXY-XXXYII. 1872-4. 8°. Annuaire. 1874. 8°. Centieme anniversaire de fondation. 1872. 2 vols. 8°. Observatoire Royal. Annales. T. XXI, XXII. 1872-3. 4°. Observations des phenomenes periodiques. 1872. 4°. Notices extraits de r Annuaire pour 1874. 16°. Quetelet, A. Congres international de statistique. 1873. 4°. Quetelet, E. La comete de Coggia. 8°. pp. 10. Quetelet, E. Rapport sur I'areography de M. Terby. 8 \ p. 6. Societe Entomologique de Belgique. Annales. T. I-XIV, XYI. 1857-1873. 8°. Compte-rendu. Ser. IL No. 18. 1875. 8°. Societe Geologique de Belgique. Annales. T. I. 1874. 8°. Buenos Ayres. — Academia Nacional de Ciencins Exactas. Boletin. Entrega I. 1874. 8°. Calcutta.— A9ia<ic Society of Bengal. Journal. 1873. I. 2-4, II. 3-4; 1874; 1875, 1., II. 1-3. 8°. Proceedings. 1873. v-x, 1874, 1875. 8°. Carlsruhe. — Polytechnische Schule. Programm. 1875-6. 8°. Riffel, A. Ueber die anatomischen und physiologischen Eigenschaften der ausseren Haut. Tubingen, 1875. 8°. Additiuns to the Lihrary. vii Catania. — Accademia Gioenia di Scienze Naturali. Atti. Ser. III. T. VII, VIII. 1872-3. 4°. Carta geologica della citta di Catania e dintonii. Per Car- melo Scinto-Patti. 8 Tavole. Chemnitz. — Naturwibsenschaftlidie Gesellschaft. Bericht IV. 1873. 8°. Cherbourg. — Societe Nationale des Sciences Naturelks. Memoires. T. XVII, XVIII. 1873. 8". Catalogue de la bibliotheque. 2"* partie, 1" livr. 1873. 8°. Chur. — Naturfurschende Gesellschaft Graubiindens. Jahresbericht. Neue Folge. XV -XVII. 18G9-72. 8°. Copenhagen. — Kongelige Danske Videnskaberaes Selskab. Oversigt over Forliandlin- ger. 1873, 1874, 1875, i. 8°. Danzig. — Naturforschende Gesellschaft SchrLfteu. Neue Folge. Bd. III. 2-3. 1873-4. 8°. Dijon. — Academie des Sciences, Arts et Belles Lettres. Memoires. III. Ser. T. I. 1871-3. 8°. DORPAT. — Gelehfte Estnische Gesellscliaft. Verhaudluiigeu. Bd. VIII. 2. 1875. 8°. Sitzungstaericht. 1874. 8°. Naturforscher Gesellschaft. Sitzung.sberichte. Bd. III. 1-6, IV. 1. 1869-75. 8°. Archiv fiir die Naturkunde Liv-, Ehsl- uud Kurland.s. I. Ser. Bd. V. 1-4, VI, VII, 1-4. 1870-5. II. Ser. Bd. V, VII. 1-2. 1867-75. 8 . Dresden. — Kais. Leopold.- Carolin. Deutsche Akademie der Naturforscher. Leopoldiiia. HeftVII-X. 1871-4. 4°. Naturwissenschaftliche Gesellschaft Isis. Sitzungsberichte. 1873, Apr. -Dec, 1874, Jan.-Sept., 1875, Jau.-Dec. 8°. Verein fur Erdkunde. Jahresbericht. X, XI, XII. 1874-5. 8°. Dublin. — Royal Irish Academy. Transaetious. Vol. XXIV; Antiquities, Ft. ix; Vol. XXV; Science, Pt. i-xx. 1872-5. 4^. Proceedings. Vol. X. 4; Series II. Vol. II. 1-3. 1870-5. 8°. Edinburgh. — Geological Society. Transactions. Vol. II. 3. 1874. 8°. Emden. — Naktrfwschende Gesellschaft. Kleine Schriiten. XVII. Hannover, 1875. 4°. Jahresbericht. LIX, LX. 1873-4. 8°. Erfurt. — Konigl. Akademie gemeinniitziger Wissenschaften. Jalirbuch. Neue Folge. Heft. VII. 1873. 8°. Falmouth. — Royal Cornwall Polytechnic Society. Forty-second Annual Report. 1874. 8°. FiRENZE. — R. Comitato Geohgico d'ltalia. Bolletino. 1873, 1874, 1875. i-iv. 8°. Frankfurt a. M. — Neue Zoologische Gesellschaft. Der Zoologische Garten. Jahrg. XIV. 7-12, XV, XVI. 1-6. 1873-5. 8°. Freiburg i. B. — NatMrforschende Gesellschaft. Berichte. Bd. VI. 1-2. 1873. 8'. Geneve. — Institut National Genevois. Bulletin. T. XX. 1875. 8". Societe de Physique et d^ Histoire Naturelle. Memoires. T. XXIII. XXIV. 1. 1873-5. 4°. Glasgow. — Philosophical Society. Proceedings. Vol. IX. 1873-5. 8°. Gorlitz. — Naturforschende Gesellschaft. Abhandhmgen. Bd. XV. 1875. 8°. Goteborg. — Kongl. Vetenskaps och Vitterhets-Samhallr'. Handlingar. Ny Tidsfoljd. Haftet XII-XIV. 1873-4. 8°. Halle. — Naturfarschende Gesellschaft. Abhandhmgen. Bd. XII. 3-4, XIII. 2. 4°. Bericht. 1873, 1874. 4". Nalurwissenschaftlicher Verein fiir Sachsen und Thilringen. Zeitschrift der gesammten Naturwissenschaften. Bd. VII-X. Berlin, 1873-4. 8°. viii Additions lo the Lihrary. Hamburg. — Xaturwissemchafllicher Verein. Abhandlungon. Bd. V. -i, VI. 1. 1873. 4". Hannover. — Naturhislorische Gesellschaft. Jahresbericht. XXII-XXIV. 1872-4. 8°. Harlem. — Musee Teyler. Archives. Vol. III. 3. 1873. 8°. Heidelberg. — NaturMstorisch-Medecinischer Verein. Verhandlungen. Neue Folge. Bd. I. 1, 3. 1874-6. 8°. Helsingfors. — Societas Sdentiarum Fennica. Ofversigt af Forhandlingar. XIV- XVI. 1871-4. 8'. Bidrag till Kannedom af Finlands Natur och Folk. Haft. XVIII, XIX, XXI, XXII, XXIII. 1871-3. 8\ Hermannstadt. — Siehenhurgischer Verein. Verhandlungen. Jahrg. XXV. 1875. 8". Hobart Town. — Royal Society of Tasmania. Monthly Notices for 1872. 8". KoNiGSBERG. — Konigl. physikalisch- okonomische Gesellschaft. Schriften. Jahrg. XIV- XV. 1873-4. 4°. Krakau. — K.k. Ste7-nivarte. Materialy do Kllimatografii Galicyi. Rok 1872-1874. 8'. Lausanne. — Societe Vaudoise des Sciences Naturelles. Bulletin. II. Ser. No. 64-65, 71-75. 1870-1876. 8°. Leiden. — Sternwarte. Annalen. Bd. IV. Haag, 1875. 4^ Leipzig. — Astronomische Gesellschaft. Vierteljahrsschrift. Jahrg. VIII. 3-4, IX, X. 1-3. 1873-5. 8°. Publication XIII. 1874. 4". Liege. — Societe Royalf. des Sciences. Memoires. II. Ser. T. IV, V. 1873-4. 8". LiNZ. — Handels- und Geiuerbekammer. Bericht. 1870, 1871, 1872. 8°. Bericht iiber die Lage und Bediirfnisse des Kleingewerbes in Oberosterreich. 1872. 8 . Liverpool. — Literary and Philosophical Society. Proceedings. No. XXVII-XXIX. 1872-5. 8°. London. — Mathematical Society. Proceedings. No. 62-86. 1873-6. 8°. Luxembourg. — Institut Royal Grand-Ducal. Publications. T. XIII. Section des Sci. Nat. et Mathemat. T. XIV, XV. 1873-5. 8°. Observations Meteo- rologiques faites a Luxembourg par F. Renter. Vol. II. 1874. 8 . Lyon. — Academie des Sciences, Belles-Lettres et Arts. Memoires. Classe des Sciences. T. XX. 1873-4. 8=. Manchester. — Literary and Philosophical Society. Memoirs. III. Ser. Vol. IV. 1871. 8°. Proceedings. A^ol. VIII-XII. 1869-73. 8°. Scientific Students Association. Annual Report. 1872,1873. 8". Melbourne. — Royal Society of Victoria. Transactions and Proceedings. Vol. X, XI. 1874. 8°. Metz. — Academie. Memoires. Annee L-LV. 1868-74. 8°. Tables generales. 1819-1871. 8\ Mexico. — Sociedad de Geografia y Estadistica. Boletin. Ill Epocha. T. II. 5-6. 1875. 8°. MiLANO. — Reale Instituio Lomhardo. Rendiconti. Serie II. Vol. VI. 6-20. VII. 1-16. 1873-4. 8 . Reale Osservatorio di Brera. Publicazione. No. II, IV, V, VII-X. 1873-5. 4°. Societd Italiana di Scienze Xaturali. Atti. Vol. XV. 3-5. XVI, XVII. 1-3. 1872-5. 8°. . MONTPELLiER. — Academie des Sciences et des Lettres, Memoires. Lettres, T. V. 4 ; Sci- ences, T. VIIL 2; Medecine, T. IV. 6. 1870-2. 4°. Moscow, — Societe Imperiale des Naturalistes. Nouveaux Memoires, T- XIII. 4, 1874, 4°. Bulletin. 1873, ii-iv, 1874, i-iv. 8°, Additions to the Library. IX MUNCHEN. — Konkjl. Bayerische Akademie der Wissenschaften. 8itziingsberifhte der phi- losoph.- philolog. und histor. Classe. 1872. iv-v, 1873, 1874. i-iii. 8 . Sitzungsberichte der mathemat.-physikal. Classe. 1872. iii, 1873. 1874. i-ii. 8". Bietz, W. Der Antheil der konigl. Bayer. Akademie an der Eutwick- lung der Electricitatslehre. Miinchen, 1873. 4". Bisc'hoff, T. L. W. von. Ueber den Einfluss des Freih. Justus v. Liebig auf die Entwicklung der Physiologie. Miinchen, 1874. 4°. DoUinger, J. von. Rede, 25 Juli, 1873. Miinchen, 1874. 8". Pettenkofer, Max von. Dr. Justus Freih. von Liebig. zum Gedaclitniss. Miinchen, 1874. 4°. Prantl, K. von. Gedachtnissrede auf F. A. Trendelenburg. Miinchen, 1873. 4°. Vogel, A. Justus Freili. von Liebig als Begriinder der Agrikultur- Chemie. Miinchen, 1874. 4. Sternwarte. Annalen. Bd. XIX. 1873. 8°. Supplementbd. XIII. 1874. 8°. Landesioirtlischaftlicher Verein in Boyern. Haus- nnd Landwirthschafts Kalender. 1874. 4°. Napoli. — Societd Reale di NapoU. Accademia delle Scienze Fisiclie e Matematiche. Atti. Vol. V. 1873. 4° Rendiconto. Anno IX-XI. 1870-72. 4°. Neu-Brandenburg. — Verein dm- Freunde der Naturgeschichte in Meckknbvjy. Archiv. Jahrg. XXVII-XXIX. 1873-5. 8°. Neuchatel. — Societe des Sciences NatMrelles. Bulletin. T. IX, X. 1-2. 1871-5. 8". Offenbach A. M. — Verein fiir Ndturkunde. Bericht XIII, XIY. 1871-3. 8°. Paris. — Societe d'Acclimatation. Bulletin Mensuel. II Ser. T. X. 6-1 1. Ill Ser. T. I. 2-12, II, IIL 1. 2. 1873-6. 8°. Societe Geologique de f ranee. Bulletin. Ill Ser. T. I. 1-5, II. 1-5, 7, and Tables, III. 1-2, 4-5. S, IV. 1. 1872-6. 8'. Societe Americaine. Annuaire. 1873. 8°. Peag. — Konigl. hohmische Akademie der Wissenschaften. Abhandlungen. Sechste Folge. Bd. VI, VIL 1873-4. 4°. Sitzungsberichte. 1872. ii, 1873, 1874. 8°. K. k. Sternwarte. Astronomische. magnetische und meteorologische Beobachtungen. 1873, 1874. 4°. PuLKOWA. — Nicolai Hauptsternwarte. Jahresbericht. 1871-2, 1872-3. 8°. Dollen, "W. Die Zeitbestimnumg vermittelst des tragbaren Durchgangs- instruments im Verticale des Polarsterns. Zweite Abhandluug. St. Petersburg, 1874. 4^. Quebec. — Literary and Historical Society. Transactions. New Series. Part X, XI. 1873-5. 8°. Regensburg. — Zoologisch- mineralogischer Verein. Abhandlungen. Heft X. Miinchen, 1875. 8°. Correspondenz- Blatt. Jahrg. XXVIL XXVIII. 1873-4.8". Historischer Verein vom Oberpfalz und Regenshurg. Verhandhmgen. Bd. XXIX, XXX. Verzeichniss iiber Bd. I-XXX. 1874. 8°. Riga. — Naturforscher Verein. Correspondenzblatt. Jahrg. XX. 1874. 8°. St, G ALLEN, —Naturwissenschaftliche Gesellschaft, Bericht. 1872-3, S'', X Additions to tlie Lihvary. Santiago. — Uniuersidad de Ckik. Annies. T. XXVIII-XLIY. 1866-73. 8". Memorias preseatados al Congreso Nacional de 1874, viz. : Memoria de Relaciones Esteriores e de Colonizacion ; del Interior ; de Justicia, Culto e Instruccion Publica ; de Guerra ; de Marina. Santiago and Valparaiso. 1874. 5 vols. 8°. Anuario de la Oficina Central Meteorolojica de Santiago de Chile. 1871-2. 8°. Briseno, R. Estadistica Bibliogralica de la Literatura Chiliena. Santiago. 1862. 4'. Doraeyko, D. Iguacio. Quarto Apendice al Reino Mineral de Chile. Santiago, 1874. 8°. Varas, J. A- Colonizacion de Llanquihue, Valdivia, Arauco. Santiago, 1872. 8°. St. Petersburg. — Jardin Imperial de Botanique. Trudi 1. 2, II, III. 1872-4. 8'. Schiveizerische Naturforschende Gesellscha/t. Verhandhmgeu in SchafEhausen, 1873. Jahresversammlung LVI. 8°. Stockholm. — Kowjl. Svenska Vetenskaps Akademien. Handlingar. Ny Foljd. Bd. IX. 2, X, XII. 1870-3. 4°. Bihang till Handlingar. Bd. I, II. 1872-5. 8°. Ofversigt. Arg. XXVIII-XXXI. 1871-4. 8°. Meteorologiska Jagttagelser. Bd. XII-XIV. 1870-2. 4°. Minnesteckning ofver J. A. von Hartniansdorff. 1872. 8°. Minnesteckning ofver Hans Jarta. 1874. 8°. Stuttgart. — Verein fiir vaterliindische Naturkunde in Wilrttemberg. Jahreshefte. Jahrg. XXX, XXXI. 1874-5 8°. Sydney. — Government Observatory. Results of Meteorological Observations. 1872, 1873. 8°. Toronto. Magnetical Observatory. Monthly Meteorological Register. 1873-4. 8°. General Meteorological Register. 1873-5. 8°. Abstracts and Results of Meteorological Observations, 1841-71. 1875. 8°. Third Report of the Meteorological Office of the Dominion of Canada. 1873. 8". Upsala. — 7i'e.oi« Societas Scientiarum. Nova Acta. Ser. Ill, T. VIII. 2. 1873°. 4°. Bulletin Meteorologique Mensuel. IV, V. 1-6. 1872-3. 4°. WiEN. — Kaiserliche Akademie der Wissenschaften. Sitzungsberichte. Math.- naturwiss. Olasse. Abtheil. i, ii, Bd. LXIII, LXIV. Abtheil. i, Bd. LXV-LXXI. 1871-5. 8^ Wex, G. Ueber die Wasserabuahme in den Quellen, Fliissen und Stromen. Wien, 1878. 4°. K. k. geologisclie Reichsanstalt. Abhandlungen. Bd. V, VI, VII, 1-3, VIII. 1. 1871-5. 4°. Jahrbuch. Bd. XIX. 4, XX-XXV. 1869-75. 8°. Verhandlungen. Jahrg. 1869. No. 14^-18, 1870-3, 1874. No. 1-13, 16-18, 1875. 8". General Register zu Jahrbuch XI-XX und Ver- handlungen 1860-1870. 8°. Hauer, F. v. Zur Erinnerung an Wilhelm Haidinger. 8°. A'. A;, zoologisch- botanische Gesellschaft. Verhandlungen. Bd. XXIII, XXIV. 1873-4. 8". Wiesbaden. — Nassauischer Verein fiir Naturkunde. Jahrbiicher. Jahrg. XXV- XXVIII. 1871-~1. 8°. Addiiiovs in the Lihrcvn/. xi WuRZBURG. — Physihiliscli- rnedicinifiche Gesellscha/f. Sitzimgsberichte. 1868-74. 8"^. Festschrift ziir Feier des fiinfundzwanzigjahriges Bestehens der Gresell- schaft. 1875. 4°. Kolliker. A. Festrede, 8. December, 1874. 8°. Zurich. — Natwforschende Gesellschaft. Yierteljahrsshrift. Jahrg. XIV-XVII. 1869 -72. 8°. Agardh, J. U. Till Algernes Systematik. Xya Bidrag. Lund, 1872. 4". Frmn the Author. Galle, J. G. Ueber eine Bestimung der Sonnen- ParaUaxe aus correspondirenden Be- obachtungen des Planeten Flora, 187.'?. Breslau, 1875. 8'. From' the Author. Haughton, R. Principles of Animal Mechanics. Second Edition. Lond., 1873. 8". From the Author. Macodo, J. M. de. Notions de chorographie du Brezil. Leipzig, l87o. 8^. Schiner, J. R. Diptera Austrica. I. Wien, 1854. iSeparatabdruck naturwiss. Abhandlungen aus den Schriften des zoolog.-botanischen Vereins, in Wien. Wien, 1856. 8. From Dr. V. Ploson. Meunier, S. Cours de geologic compareo. Paris, 1874. 8 . From the' Author. Morren, fi. L'horticulture a rF,xposition Universelle de Paris de 1867. Bruxelles, 1870. 8\ Rapport seculaire sur les travaux de botanique et de physiologic vege- tale. Bruxelles. 8^. Eloge de Jean- Theodore Lacordaire. Liege. 1870. 8". From the Author. Newberry, J. S. The Surface Geology of Ohio. (Jolumbos, 1874. 8^. Tlie Structure and Relations of the Dinicthys. Witli two charts. Columbus, 1875. 8~. From the Author. Saussure, H. de. Memoires pour servir a I'histoire naturelle de Mexique, des Antilles et des Etats Unis. Premiere livraison. Crustacees. Geneve, 1858. 8" From the Author. Winchell, N. H. Geological and Natural History Survey of Minnesota. Second and third Annual Reports. 1873. 1874. St. Paul, 1874-5. 8°. From the Author. I. Report ox the Dr edgings in the region of St. George's Banks, in 1872.* By S. I. Smith and O. Harger. [Published by permission of tlie Superintendent of the U. S. Coast Survey.] During the summer of 1872, a series of dredgings was carried on by the authors in the neighhorliood of St. George's Banks. The work was undertaken at the instance of Professor Baird, United States Commissioner of Fish and Fisheries, and carried on, through the cooperation of the Coast Survey, from the steamer Bache, on board of which accommodations were furnished for two persons, with the necessary apparatus. On board the steamer we were received and treated throughout with the utmost courtesy by Commander J. A. Howell, and the other officers of the vessel. Lieutenants Jaques, Hagerman, Jacob and Rush ; and although the dredging was carried on in connection with the special hydrographic work of the Coast Survey, all these gentlemen manifested a degree of interest in our work equal to that which they felt in their own. On account of the lateness of the season at which operations were begun, the weather was most of the time cpiite unfavorable for dredg- ing, so that the number of hauls made with the dredge was much smaller than had been expected, and no opportunities were afforded for using the large traAvl or the rake dredges which had been pro- vided, with the rest of the outfit for the natural history department of the expedition, by the United States Fish Commissioner. Still, the collections which were made from these comparatively few dredgings have proved rich and very important, giving nearly the only informa- tion which we possess of the character of the fauna of the fishing banks, and adding very largely to the knowledge of the distribution, both geographical and bathymetrical, of the marine animals of our northern coast. * The text of this report was written and presented to the Superintendent of the Coast Survey, very nearly in its present form, in December, 1872, but its publication has been unavoidably delayed until the present. The figures illustrating some of the species mentioned have been added since the report was first prepared. Trans. Connecticut Acad., Vol, III. 1 xily, 1874. 2 Smith and Harger — St. George's Banks Dredgings. After we were obliged, late in September, to leave the expedition. Prof. A. S, Packard, Jr., and Mr. Caleb Cooke, of Salem, Mass., went in the steamer on another trip, which was mainly devoted to dredg- ing. On this trip a number of successful hauls were made at differ- ent localities along the northern extremity of George's Bank, in 40 to 150 fathoms. The region visited on this trip was quite distant from any of the localities examined by us, and the bottom, in the deeper dredgings, was of an entirely different character, so that the collec- tions made by Prof Packard and Mr. Cooke contain many species not found by us, and add very greatly to the value of the results. We wish specially to acknowledge the assistance rendered us in the preparation of this report by Professor Verrill, who has identified all the worms mentioned, and the more difficult mollusks and radiates. The following table will facilitate references to the localities at which the dredgings were made. The letters in the first column are the same as those used by Professor Verrill in his papers in the American Journal of Science. When more than one haul of the dredge was made at any of the localities, the number of hauls is indi- cated in parenthesis. station. N. Lat. W. Long. Depth in filth. Nature of bottom. Temperature. | Air. Surface. Bottom.* a 41 40 68 10 25 Soft sand. i 30 a il c 41 25 66 45 28 Coarse saud. d U il 66 24-8 50 Sand and shells. 66° 62° 45° e " " 65 58-3 60 Shells and sand. 61 58 58 f u a 65 50-3 65 Dead shells. 64 60 55 g 11 a 65 42-3 430 Sand, gravel and stones. 66 65 51 h 42 56-5 64 51-3 45 Gravel and stones. 64 61 36 i 42 44 64 36 60 Gravel, stones and sponges. 62 62 J 0(3) 20 110 Mud and fine sand. Soft mud and sand. 56 49 42 5 67 49 P 42 3 " " 85 a u a a 50 49 q(^) 42 0 67 42 45 Coarse sand. r 42 3 67 31 40 a a 5(2) 42 11 67 17 150 Soft sandy mud. 52 52 The first dredgings were made on the evening of August 29, to the west of George's Shoal, about latitude 41° 40' north, longitude 68° 10' * Very little confidence can be placed in these bottom temperatures, as the Miller- Casella thermometers used did not give uniform results. Most of the temperatures are manifestly much too high, Smith and Harger — St. George's Banks. Dredgings. 3 west. The first haul, («), in 25 fathoms, soft sandy bottom, gave the following species : Crustacea, * Eupagurus Bernhardiis Brandt ; a1>undant. Crangon 'oulgarls Fabricins ; abundant. Conilera poUta Harger (Stimpson). Epelys montosus Smith (Stimpson). Balanus porcatus Costa ; common. Annelida. (Jistenides gramdata Malmgren. TURBELLARIA. Meckelia lurida (?) Verrill. MOLLUSCA. JBela turricuki (Montagu). B. harptdaria H. and A. Adams (Couthouy). Buccinwn tindatum Linne ; very large and abundant. Nejitunea pygmoia H. and A. Adams (Gould) ; abundant. Tritia trlvittata H. and A. Adams (Say) ; abundant. Lunatia heros H. and A. Adams (Say). L. hnma.cxdata H. and A. Adams (Totten). Crepidida plana Say {iingniformis Stimpnon) ; several, living. Ensatella Americana Verrill (Gould). Modiolaria nigra Loven (Gray). Radiata. Ee/iinarachnius parma Gray; very abundant. Sydractinia polyclina Agassiz. At the second haul (^), in '^(^ fathoms, the bottom was of the same character, but a greater variety of species was obtained. Crustacea, Cancer irroratus Say, young ; common. Eupagurus Bernhardus Brandt; a)>undant. E. pubescens Brandt ; common. Crangon vulgaris Fabricins ; abundant, / Pandahis annulicornis Leach ; common. Stenothoe peltata Smith, sp. nov. [Plate III, figures 5-8.] Fhotis (?) sp. Ampelisca sp. Xenoclea megachir Smith, sp. nov, [Plate HI, figures 1-4.] Vnciola irrorata Say. Bulichta sp. 4 Smith and Harger—St George's Banks Bredgings. Annelida. Aphrodita aculeata Linne. Clymenella torquata Verrill ; tubes only. Tu KBELLARIA. Meckelia lurida (?) Verrill. M. ingens (?) Leidy. MOLLUSCA. Bela turricula (Montagu). B. harpidaria H. and A. Adams (Couthouy). Adtnete viridula (O. Fabricius). Buccimtm undatum Linne ; large and abundant. JSFeptunea pygmma H. and A. Adams (Gould) ; large and common. Lunatia heros H. and A. Adams (Say). " " var. triseriata (Say). (Jrepidula plana Say {xinguifornds Stimpson). Siliqua costata H. and A. Adams (Say). Yoldia Umatilla Stimpson (Say). Modiolaria nigra Loven (Gray). Farrella familiaris Smitt (Gros) ; abundant. Gemellaria loricata Busk (Linne). Radiata. Echinarachnius parma Gray ; very abundant. Asterias vulgaris Stimpson. Campanidaria verticillata Lamarck (Linne). Sertularia cupressina Linne. 8. latiusculaf Stimpson. Hydrallmania falcata Hincks (Linne) ; abundant. Five successful hauls were made on the line of soundings running east from George's Bank, on the parallel of 41° 25' north latitude, to 63° 20' west longitude. The first of these hauls (c), beginning at the western end of the line, was in about longitude 66° 45' west, from 28 fathoms, coarse sandy bottom, September 16. Here the following species occixi-red. Crustacea. Cancer irroratus Say, young ; abundant. Eupagurus Bernhardus Brandt. Crangon vulgaris Fabricius ; common. Pandalus annulicornis Leach. Ampelisca sp. Annelida. Nereis pelagica Linne. Smith and Harger — St. George's Banks Dredgings. 5 TURBELLARIA. Meekelia lurida (?) Verrill. MOLLUSCA, Bela harpularia H. and A. Adams (Coutliouy). Tritia trivittata H. and A. Adams (Say) ; common, Lunatia heros H. and A. Adams (Say), variety triseriata ; common. L. imrnaculata H. and A. Adams (Totten). Crepidula fornicata Lamarck (Linne) ; one dead specimen. Scalaria Groenlandica Sowerby. Clidiophora trilineata Carpenter. Mactra solidissima Chemnitz (Gray). Astarte castanea Say. Crenella glandula H. and A. Adams (Totten). Ostrea Virginica Lister ; only dead sjjecimens. Glandula arenicola Verrill ; abundant. Radiata. Strongylocentrotus Drdhachiensis A. Agassiz. Echinarachnius 2)arma Gray. At the second haul {d), longitude 66° 24-8' west, 50 fathoms, sandy and shelly bottom, August 31, the following species occuiTed. Pycnogonida. Nymphon grossipes Kroyer. Crustacea. Cancer irroratus Say ; young. Ilyas coarctatus Leach ; abundant. Eupagurus Bernhardtis Brandt ; common. E. Kroyerii Stimpson; common. E. ptnhescens Brandt ; common. Pandalus annulicornis Leach ; common. Vetumnus serratiis Goes. Melita dentata Boeck ( Gammariis purpuratus Stimpson). 3Icera Dance, Bate (Stimpson) ; common. Cerapus rubricornis Stimpson ; common. Podocerus nitidus Stimpson. Unciola irrorata Say ; common. Palanus porcatus Costa; common. AlSTNELIDA. Aphrodita aculeata Linne. Harmothoe imhricata Malmgren (Linne). Phyllodoce catenida Verrill. [Plate IV, figure 3.] 6 Smith and Harger — St. George's Banks JDredgings. Nereis pelagica Linne ; abundant. Cistenides gramdata Malmgren. Thelepus cineinnatus Malmgren (Fabricius). Potamilla ocidifera Verrill (Leidy). P. neglecta Malmgren. Spirorbis nautiloidesf Lamarck. [Plate IV, figure 4.] Gephyrea, PJiascolosoma cmrnentariuin Verrill. MOLLUSCA. Bela turricula (Montagu). jB. harpidaria H. and A. Adams (Couthouy). B. pleurotomaria H. and A. Adams (Couthouy). B. decussata (Couthouy). Neptunea curta Verrill (Jeffreys sp. ; Fasus Islandicus Gould). N. decenicostata (Say). N. pygmo&a H. and A. Adams (Gould). lAmatia immaculata PI. and A. Adams (Totten). Natica clausa Broderip and Sowerby ; common. Amauropsis helicoides Stimpson ; rare. Crepidida plana Say [unguiformis Stimpson). Acirsa horealis Mon^h (Beck). Margarita ohscura Gould (Couthouy). Hanleia niendicaria Carpenter (Mighels and Adams). Entalis striolata Stimpson. [Plate I, figure 3.] ^olis sp. Thracia trvncata Mighels and Adams. Cyprina Islandica Lamarck (Linne). Cardium pinnulatum Conrad ; abundant. Cyclocardia borealis Coni-ad ; common. Astarte castanea Say. A. quadrans Gould. Modiolaria discors Beck (Linne). Pecten Islandicus Chemnitz (Mtlller). Anomia aculeata Gmelin ; abundant. " " smooth variety. Boltenia clavata Stimpson. Pera crystallina Verrill (Moller) ; young. [Plate VIII, figure 1.] Amaroeciuni sp. Cettidaria ternata Johnston (Busk). " " var, duplex Smitt. Caberea Ellisii Smitt (Fleming). Smith and Marger—St. George's Banks Dredgings. 7 Bugula Murrayana Busk (Bean) ; abundant. Cellaria Jistulosa Liune. Biscopora Skenei Smitt. Cellepora scabra Smitt (Fabricius). C. ramulosa Linne, var. ; with the two last species abundant on hydroid stems. Radiata. Strongylocentrotus Brohachiensis A. Agassiz. Cribrella sanguinoknta Liitken. Beptasterias compta Verrill. Ophiopholis aculeata Gray ; common. Amphipholis elegans Ljungman. Ophioglypha robust a Lyman. Hydracthiia polyclina Agassiz. Eudendrium ramosiwi Ehrenberg. E. capillar e Alder (?) Tubularia indivisa Linne. Gonothyrma Boveni AUman. Campamdaria verticdlata Lamarck (Linne) ; common. * C. Hincksii Alder. G. voliibilis Alder (Linne). Bafoea dumosa Sars (Fleming) ; abundant on Bryozoa. B. gracillima G. O. Sars (Alder) ; with last, common. Galycella syringa Hincks (Linne) ; common. Guspidella humilis Hincks. Haleciuyyi Beanii Johnston. H. tenellum Hincks. Sertidarella polyzonias Gray, var. gigantea Hincks; common. S. triciispidata Hincks (Alder) ; common. Biphasia fallax Agassiz (Johnston) ; abundant. Sertidaria cupressina Linne ; common. S. latiuscula Stimpson. S. abietina Linne. Hydrallmania falcata Hincks ; very abundant. Urticina crassicornis Ehrenberg. Sponges. ThecopJiora ihla Wyville Thompson. [Plate VII, figure 1.] Other undetermined species. At the third haul (e), longitude 65" 58-3', 60 fathoms, shelly and sandy bottom, September 16, the following were found: 8 Smith and Harger — St. George's BcmJcs Bredgings. Crustacea. Hyas coarctatus Leach ; abundant. Eupagurus Kroyerii Stimpson ; abundant. E. pubescens Brandt ; abundant. Sabinea septemcarinata Owen ; one specimen. Pandalus annulicornis Leach ; common. Paramphithoe pulchella Bruzelius (Kroyer). Melita dentata Boeck {Gammarus purpuratas Stmipson). Podoceriis nitidus Stimpson. Caprella sp. Balanus porcatus Costa ; common. Annelida. Harmothoe imbricata Malmgren. Bhynchobolus capitatus Verrill (Oersted sp., not of Claparede). Thelepus cincinnatas Malmgren (Fabricius). Spirorbis nautiloidesf Lamarck. [Plate IV, ifigure 4.] Gephyrea. Phascolosoma cmmentariutn Verrill. Mollusca. Bela molacea (Mighels and Adams). B, harpularia H, and A. Adams (Couthouy). Neptunea curta Verrill (Jeftreys sp. ; Fiisus Islandicus Gould). N. decemcostata (Say). N'. pygmcea H. and A. Adams (Gould) ; common. iMuatia heros H. and A. Adams (Say), variety triseriata. Natica clausa Brodei'ip and Sowerby. Amnuropsis helicoides Stimpson ; rare. ' Stylifer Stimpsonii Verrill; parasitic on Strongylocentrotus Dr'6- bachiensis A. Agassiz. [Plate I, figure 1.] Aporrhais occidentalls Sowerby. Acirsa borealis Morch (Beck). Margarita obscura Gould (Couthouy) ; common. Diodora noachina Gray (Linne). Hanleia mendicaria Carpenter (Mighels and Adams). Cylichna alba Loven (Brown). Entalis striolata Stimpson ; common. [Plate I, figure 3.] Bendronotus arborescens Alder and Hancock. Cyprina Islandica Lamarck (Linne) ; very abundant. Cardium pinmdattivi Conrad; common. Astarte castanea Say. « A. quadrans Gould. Smith and Harger — St. George's Banks Dredgings. 9 Astarte undata Gould. [Plate I, figures 6-9.] Modiolaria corrugata Morch (Stimpson). Pecten tenuicostattis Mighels. P. Islandicus Chemnitz (Miiller). Anomia aculeata Gmelin, smooth variety ; common. Cellularia ternata Johnston (Busk) ; common. JBagula Murray ana Busk (Bean), Gellaria Jistulosa Linne. Piscopora Skenei Smitt, variety, Cellepora scahra Smitt (Fabricius) ; with the last on hydrqid stems, Radiata. Lophothurla P\ibricii Verrill. Psolus phantapus Oken. Strongylocentrotus Drobachiensis A. Agassiz ; abundant. Echinarachnius parma Gray ; abundant. Crossaster papposus Miiller and Troschel. Crihrella sanguinolenta Liltken. Hgdractinia polyclina Agassiz ; common. Eudendrium capillare Akler. E. raniosuni Ehrenberg. Tubularia indivisa Linne ; common, Campanular'ia iiertlcillata Lamarck (Linne) ; common, C. Hincksii Alder, C. voluhilis Alder. Gonothyrma hyalina Hincks. Lafoea dumosa Sars (Fleming) ; common, L. gracillinia G. O. Sars (Alder). Grammaria ahietina^ Sars. Goppinia arcta Hincks (Dalyell) ; on hydroid stems. Sertularella tricuspidata Hincks (Alder) ; abundant. Piphasia mirabilis Verrill. P. fallax Agassiz, Serlularia latiuscida Stimpson. S. cupressina Linne. Hydrallmania falcata Hincks (Linne) ; common. Epizoantlms Americanus Verrill ; coating shells inhabited by Eupa- gurus piihescens, and also on hydroid stems. [Plate VIH, fig. 2.] Urticma crassicornis Ehrenlierg. At the fourth haul (/), longitude 65° 5i)-3', 65 fathoms, the bottom composed of dead shells, September 15, midnight, the following occurred : Trans. Conn. Acad., Vol. III. 2 , July, 1873. 10 Smith and Uarger — St. George's Banks Dredgings. Crustacea, Eupagurus Kroyerii Stimpsoii. E. puhescens Brandt. Crangon vulgaris Fabricius. MOLLUSCA. Beta decussata (Couthouy). Natica clausa Broderip and Sowerby. Stylifer Stimpsonil Verrill ; parasitic on Strongylocentrottis Bro- hachiensis A. Agassiz. [Plate I, figure 1.] Acirsa borealis Morch (Beck). Margarita, ohscnra Gould (C?outhouy), variety. Mactra solidissima Chemnitz (Gray); abundant. Cyprina Islandica Lamarck (Linne) ; common. Cyclocardia borealis Conrad. Astarte undata Gould. [Plate I, figures 6-9.] Crenella glandula H. and A. Adams (Totten). Radiata. Strongylocentrotus Brobachiensis A. Agassiz ; common. Echinarachnius parma Gray ; abundant. Hydractinia polyclina Agassiz. Tubularia indivisa Linne. Campnnularia vertieillata Lamarck (Linne). Sertularella tricitspidata Hiiicks (Alder) ; common. Sertularia ciipressina Linne. Epizoanthus Americamis Verrill ; coating shells inhabited by Eupa- gurus pubescens. [Plate VIII, figure 2.] Urticina crassicornis Ehrenbei'g. The fifth haul {g) on this line was made on the evening of Septem- ber 15, to the east of the bank, in longitude 65° 42*3' west, at a depth of about 430 fathoms, on a bottom of sand, gravel, small and large stones. Here the following species occuiTed : Pycnogonida. Pycnogonum Uttorale Mtiller {pelagicum Stimpson) ; common. Ckustacea. Eupagurus Kroyerii Stimpson ; common. Pandalus annulicornis Leach ; several specimens. Thysanopoda sp, ; several specimens, perhaps not from the bottom. Unciola irrorata Say ; several specimens. Melita dentata Boeck. Scalpelhaii Stroemi Sars ; on hydroid stems. [Plate III, fig. 9.] Annelida, Nothria conchylega Malmgren (Sars) ; abundant. [Plate VII, fig. 3,] Smith and Hargei — St. George's Banks Dredgings. 11 Nephthys circinata Yerrill, sp. nov. Lximbriconereis frag His ffirsted (Mtiller). Leodice vivida Verrill [Eunice vivida Stimpson), [Plate Y, fig. 5.] Mhynchoholus capitatus Verrill (CErsted sp., not of Claparede). Gephyrea. Phascolosoma tubicola Verrill. MOLLUSCA. Beta cancellata (Mighels and Adams). Neptunea pygmma H. and A. Adams (Gould). Z/unatia Groenlandica (Mollcr). L. immaculata H. and A. Adams (Totten). Natica clausa Broderip and Sowerby. Margarita ohscura Gould (Couthouy). Diodora noachina Gray (Tjinne), variety princep)s (Mighels and Adams). Entails striolata Stimpson. [Plate I, figure 3.] Astarte lens Stimpson ; dvv^arf variety. Cryptodon obesns Verrill. [Plate I, figure 11.] Pecten pustulosiis Verrill. Vescictdaria armata Verrill ; on Sertularia argentea. Several other species of Bryozoa. Radiata. Lopthothuria squamata, Verrill. Pentaeta asshnills (Duben and Koren). Schizaster fragilis Agassiz (Duben and Koren). Strong ylocentrot as Probac/iiensis A. Agassiz ; several. Echinarac/mixs p((riiia Gray ; common. Ophioglypha Sarsii Lyman ; common. Eudendritim ramosuvi Ehrenberg (Linne). Tubidaria indivisa Linne. Campanularia virticillata Lamarck (Linne). Lafoea graclUhna G. O. Sars (Alder). Calycella producta G. O. Sars. Halecinm robustuui Vei-rill. Sertularella Gayi Hincks (Lamoroux). S. triciispidata Ilincks (Aldei-) ; with reproductive capsules. Sertularia argentea Linne, slender variety; with reproductive capsules. Epizoanthus Americanus Verrill ; upon small stones. Urticina crassicornis Erhenberg ; young specimens. U. nodosa Verrill (Fabricius) ; two large specimens. 12 Smitfi and Harger — St. George's Banks Dreclgmgs. On the line of soundings from near Cape Sable, Nova Scotia, to lati- tude 41° 25' north, longitude 63° 20' west, two successful hauls were made September 12, on Le Have Bank. The first (A), latitude 42° 56-5' north, longitude 64° 51 '3' west, 45 fathoms, gravelly and stony bot- tom, gave the following species : Crustacea. Hyas coarctatus Leach ; very abundant. Eupagurus Kroyerii Stimpson ; abundant. Hippolyte spina Leach ; several specimens. a. pusiola Kroyer. ^ Syrrhoe crenulata Goes ; a single specimen. Tiron acanthurus Lilljeborg ; one specimen. Paramphitho'e cataphracta Smith (Stimpson). Tritropis aculeata Boeck, and several undetermined Amphipods. Annelida and Gephyrea. Nychia cirrosa Malmgren (Pallas). Eunod nodosa Malmgren (Sars). JELarmothoe imhricata Malmgren (Linne). Nereis pelagica Linne ; abundant. Nothria conchylega Malmgren (Sars) ; very abundant. [Plate VII, figure 3.] Spiocluetopterus (?) ; tubes only. Cistenides granulala Malmgren, Thelepus cincinnatus Malmgren (Fabricius). Potamilla ocidifera Verrill (Leidy). P. neglecta Malmgren ; very abundant. Spirorhis valida, Verrill, sp. nov. S. hicidus Morch. Phascolosoma ccementarium Verrill. TURBELLARIA. Leptoplana ellipsoides Girard. MOLLUSCA. Bela violacea (Mighels and Adams). Admete viridula Stimpson (O. Fabricius). Buccinunn undatwm Linne ; common. Neptunea deeemcostata (Say). N. pygmoea H. and A. Adams (Gould) ; common. Trophon Gnvneri Loven ; three sjDecimens. Lunatia Groenkwdica. (Moller). Natica clausa Broderip and Sowerby ; abundant. Grepidula plana Say {unguiformis Stimpson); one alive. TVichotropis horealis Sowerby; abundant. Smith and Harger — Sf. George's £anks Dredgings. 13 Aporrhais occidentalls Sowerby. Turritella erosa Couthoiiy ; common. Scalaria Groerdandica Sowerby ; abuudant. Margarita cinerea Gould ; common. M. Groenlandica Moller {M. undulata Gould). Trachydermon album Carpenter (Montagu). Entails striolata Stimpson; abundant. [Plate I, figure 3.] Dendronotus arhoreseens Alder and Hancock. Mya, truncata Linne. Cardiian pinnulaturn Conrad ; common. Cyclocardia N'ovanglim Morse ; common. Astarte elliptica (Brown); very abundant. [Plate I, figure 10.] A. Banksii Leach; common. [Plate I, figure 12.] A. undata Gould ; common. [Plate I, figures 6-9.] Pecten Islaudicus Chemnitz (Mtiller) ; abundant. Boltenia Molteni. Cynthia carnea Verrill. Terehrattdina sep)tentrionalis (Couthouy) ; common. Mhynchonella psittacea (Gmelm). Myriozoum coaretatam Smitt (Sars) ; common. Eschara papposa Packard. Escharoides rosacea Smitt. Cellepora avicidaris Hincks. Radiata. Lophothuria Fahricii Verrill. Strongylocentrotus Drobachlensis A. Agassiz; common. Grossaster papposus Mtiller and Troschel; young. Pteraster milltaris Mtiller and Troschel. OphiophoUs acideata Gray ; common. Ophioglypha Sarsii Lyman. 0. robusta Lyman. Clytia Johnstoni Hincks (Alder). Hydractinia polyclina Agassiz ; abundant. Eudendriunti cappilare Alder. Tubidaria indivisa Linn^ ; common. Cainpanularia verticillata Lamarck (Linn6). C. IIi7icksii Alder. Lafoea graclllhna G. O. Sars. Calycella syringa Hincks (Linne). Sertularella tricuspidata Hincks (Alder); common. S. polyzonias Gray, variety gigantea Hincks. Thuiaria articulata Fleming (Pallas). Urticina crassicornis Ehrenberg ; abundant. 14 Smith and Jffarger — St. George's Banks Dredgings. At the second haul [i), latitude 41° 44' north, longitude 64° 36' west, 60 fathoms, coarse gravel, stones, and sponges, the following occurred : Pycnogonida. Nymphon grossipes Kroyer, Crustacea. Hyas coarctatus Leach. H. araneus Leach. Eupagurus Kroyerii Stimpsou ; abundant. Sabinea septemcarinata Owen; two specimens. Tritropis aculeata Boeck. Acanthozone cuspidata Boeck. Annelida. Eunoa nodosa Malragren (Sars). Harmothoe imhricata Malmgren (Linne). Lagisca rarispina Malmgren (Sars). Nothria conchylega Malmgren (Sars) ; abundant. [Plate VII, fig. 3.] Thelepiis cinclnnatus Malmgren (Fabricius). I*otamUla neglecta M'dlmgYQn ; abundant. Spirorhis valida VeiTill. MOLLUSCA. Adrnete viridula Stimpson (O. Fabricius) ; common, Trophon Gunneri Loven. Natica clausa Broderip and Sowerby. Aporrhais occidentalis Sowei'by. Turritella reticulata Mighels and Adams. Margarita cinerea Gould. Hanleia mendicaria Carpenter (JVlighels and Adams). Trachydermon album Carpenter (Montagu). Entalis striolata Stimpson ; abundant. uEolis rufibranc.hialis Alder and Hancock (?). Cardium pinmdatum Conrad. Terebratulina septentrionalis (Couthouy) ; common. Atnarcecium glabrum Verrill. Discopora Skefiei Smitt. Radiata. Gribrella sanguinolenta Liitken. Ophiopholis aculeata Gray ; abundant. Ophioglypha Sarsii Lyman. Lafoea gracillima G. O. Sars. Sertularella tricuspidata Hincks. Smith and Harger — St. George's Bmiks Dredgings. 15 Sertularella polyzoiiias Gray, variety gigantea Hincks. Hydrallmania falcata Hincks, var. tenerrima (Stimpson). Aglaophenia myriophylhdn Laraoroux (Linne), Granimaria abietina Sars. Many species of sponges were also obtained, but most of them are as yet undetermined. Among them is Thecophora ibla W. Thompson. On leaving Halifax, N. S., September 11, one haul (j/) was made just off Chebucto Head, in 20 fathoms, soft mud and fine sand with decaying seaweed. Here the following were found: Crustacea. Hyas araneus Leach ; common. Eupagtiriis pubescens Brandt. Diastylis quadrispinosa G. O. Sars ; common. D. sculpta G. O. Sars. Halii-ages fidvocinctus Boeck (Sars). Gammarus ornatus Edwards ; perhaps from floating sea-weed. Ampelisca sp. ; common. MonoGxdodes borealis Boeck. Annelida. Harmothoe imbricata Malmgren (Linn6). Goniada maculata OErsted. Brada sp. Cistenides granulata Malmgren. Mollusca. Aporrhais occidentcdis Sowerby; common. Turritella reticulata Mighels and Adams ; common. Margarita varicosa IMighels and Adams ; common. M. obscura Gould (C'outhouy), variety. Thracia niyopsis Beck. Macoma proxima (Gray). Astarte elliptica (Brown). [Plate I, figure 10.] Anomia aculeata Gmelin. Terebratidina septentrioualis (Couthouy). Rhynchonella psittacea (Gmelin). Grisia eburnea Lamoroux (Linne). [Plate II, figures 3-4.] F lustra papyrea (Pallas). Radiata. Ophiopholis aculeata Gray. Ophioglypha robusta Lyman. Manania auricula Clark (?) Hydrallmania falcata Hincks (Linne). 16 Smith and Han/er — St. George's Hanks Dredgings. The dredging while Dr. Packard and Mr. Cooke were on board the Bache was at five localities. First (o), just on the northwestern bor- der of George's Bank, latitude 42° 5' north, longitude 67° 49' west, in 110 fathoms, three hauls were made from a bottom of soft sandy mud with a few stones, and the following collected : Crustacea. Caridion Gordonl Goes ; one small specimen. Ilarpina fusiformis (Stimpson) ; common. Stegocephalus ampulla Bell ; one large specimen. Unciola irrorata Say ; common. Anthura brachiata Stimpson. Annelida. Lmnillaf mollis G. O. Sars. Pholoe mimita Malmgren. Nephthys ciliata Malmgren (Miiller). [Plate V, figure 1.] N. ingens Stimpson. Phyllodoce sp. Eteone depressa jNIalmgren (?). Nereis pelagica Linne ; common. Lumbriconereis fragilis Oersted (Mtiller). [Plate V, figure 2.] Ninoe nigripes Verrill. [Plate V, figure 3.] Leodice vivida Verrill (Stimpson). [Plate V, figure 5.] Nothria coiwhylega Malmgren (Sars). [Plate VII, figure 3.] N. opalina Verrill ; common. [Plate VII, figure 4.] Goniada maculata CErsted. Rhynchoholus capitatus Verrill (CErsted sp., not of Claparede). Ammotrypane fimhriata Verrill. " Eamenia crassa CErsted. Trophonia aspera Verrill (Stimpson). Sternaspis fossor Stimpson. Scolecolepis cirrata Malmgren (Sars). Nbtom,asti(S latericins Sars. Ancistria capillaris Verrill. Maldane Sarsii Malmgren. Rhodine Zioveni Malmgren. Nicomache lumhrlcalis Malmgren (Fabriciusj. Axiothea catenata Malmgren (?). Praxilla pnetermissa Malmgren. P. gracilis Malmgren. P. species undetermined. Am.m,ochares assimilis Sars. [Plate V, figure 4.] Ampharete arctiea Malmgren. Smith and Harger — St. George's Banks Bredgings. V Ampharete Finrnarchira (?) Malmgren (Sars). A. gracilis Malmgren. Amphicteis Gunneri Malmgren (Sars) ; abundant. Samytha sexcirrata Malmgren (Sars). Samythella elongata Verrill. Melinna cristata Malmgren (Sars.) Amphitrite cirrata Miiller. Pista cristata Malmgren (Mtiller). [Plate IV, figure 2.] Grymma spiralis Verrill. [Plate IV, figure 1.] Terebellides Stroemi Sars. Polycirrus sp. Sahella pavonia (?) Malmgren. Fotamilla neglecta Malmgren. Protxda media Stimpson ; tubes only. [Plate VI.] P. horealis (?) Sars ; tubes only. Gephyrea. Phascolosoma cmmentarium Verrill. P. boreale Keferstein (?) P. tuhicola Verrill. Chcetoderma nitidulitm Loven, [Plate VIII, figures 3-4.] TURBELLARIA. Meckelia lurida (?) Verrill. MOLLUSCA. Pleurotoniella Packardii Verrill ; one living. Adniete viridula Stimj^son (O. Fabricius). JVeptunea pygmcea H. and A. Adams (Gould) ; common. Ringicula nitida Verrill. [Plate I, figure 2.] Lunatia Groenlandica (Mollei'). Natica clausa Broderip and Sowerby. Velutina Icevigata (Linne). Margarita citterea Gould. Lepeta, cmca Mtiller. Trachydermon alburn Carpenter (Montagu). Cyliclina alba Loven (Brown). Philine s]>. Entalis striolata Stimpson ; common. JVecera arctiea Sars. Thracia myopsis Beck. Macoma proxinta (Gray). Gardiuni pinnidatum Conrad. Astarte lens Stimpson, d^varf var. ; common. Teans. Conn. Acad., Vol. ITI. 3 August, 1874. 18 Smith and Harger — St. George's Banks Dredging s. Cryptodon Gouldii H. and A. Adams (Pliilippi). Yoldia obesa Stimpson. Leda tenuisulcata Stimpson (Couthouy) ; common. (Jrenella glandula H. and A. Adams (Totten), Pecten Ishindlcus Chemnitz (Mullei"). P. tenuicostatus Mighels. Aiioniia acideata Gmelin. Ascldiopsis complanata Verrill (Fabricius). TerebratuUna septentrionalis (Couthouy). Dtscofascigera lucernaria Sars. Cellularia sp. Bugula avicularia Busk, v a.viet j /astigiata. Radiata. Lophothuria Fabricii Verrill. Pentacta assimilis (Duben and Koren). Thyone scabra Verrill. Schizaster fragilis Agassiz (Duben and Koren) ; several. Ctenodiscus crispatus Duben and Koren. Ophioglypha affinis Lyman. 0. Sarsii Lyman. Opiopholis aculeata Gray. Ophiacantha spimdosa Mtlller and Troschel. Archaster arcticus Sars. Pennatida acideata Danielsen. Gerianthus borealis Verrill. [l^late 11, figure 5.] Second (jo), a little to the southeast of the first, latitude 42° 3', lon- gitude 67° 49', 85 fathoms, one haul from a bottom of the same char- acter as at first locality : Crustacea. Harpina. fusiformis Smith (Stimpsoji) ; common. Aknelida. Antinoe Sarsii Kinberg. Neplithys ingens Stimpson. N. circinata Verrill, sp. nov. Lmnbriconereis fragilis (Ersted (Miiller). [Plate V, figure 2.] Nothria eonchylega Malmgren (Sars). [Plate VII, figure 3.] ^4 rn mo try pane firnbriata Ve rr il 1 . JEmnenia crassa Oersted. Trophonia aspera Verrill (Stimpson). Sternaspis fossor Stimpson. Ghoetozone setosa Malmgren. Smith and Harger — St. George's Banks Dredgings. 1 9 Ancistria capillaris Verrill. Maldane Sarsii Malmgren. Mhodine Loveni Malmgren. Pr axilla praetermissa Malmgren. Anmiochares assiniilis Sars. Ampharete (irctica Malmgren. Ainphirteis Sundevalli Malmgren. Terehellides Stroetni Sars. Polycirrus sp. Gepuyrea. Phascolosoma cmmentarium Verrill. MOLLUSCA. Natica clausa Broderip and Sowerby. Scalar la Grcenlandlc<i Sowerby. Yoldia obesa Stimpson. Y. thraciformis Stim))Son (Storer). Radiata. Schizaster fragiUs Agassiz (l)uben and Koren). Edwardsia sp. Third (</), still farther to the southeast, latitude 42°, longitude 67° 42', two hauls in 45 fathoms, coarse sandy bottom : Crustacea. Hyas coarctatus Leach ; very abundant. Cancer Irroratus Say ; one young specimen. Eupagurus Bernhardus Brandt. E. Kroyeri Stimpson ; common. E. puhescens Brandt ; common. Crangon vulgaris Fabricius ; abundant. Hippolyte pusiola Kroyer. Pandalus annidicomls Leach ; common. Vertin)i7iKS serratus Goes. Paramphithoe cataphracta. Smith (Stimpson). P. pnlchella Bruzelius. Phoxus Kroyeri Stimpson. Melita dentata Boeck, Pontogeneia inermis Boeck ; one specimen. Gerapus rubricornis Stimpson. Xenoclea rnegachir Smith, sp. nov. [Plate III, figures 1-4.] JJnciola irrorata Say. Balanus porcatus Costa. 20 Smith ayxd Harger — St. George's Banks Dredgings. Annelida. Harmothoe imhricata Malmgren. JOagisca propinqua Malmgren. Eusyllis phosphorea Verrill, sp. nov. [Plate VII, figure 2.] Nereis pelagica Linne. Leodice vivida Verrill (Stimpson). [Plate V, figure 5.] Nothria conchylega Malmgren (Sars). [Plate VII, figure 3. J Am7noehares assirtdlis Sars. [Plate V, figure 4.] Amphitrite Groenlandica Malmgren. Thelepus cincinnatus Malmgren (Fal)ricius). Chone infundibidiformis Kroyer. Spirorhis nautiloides Lamarck ? MOLLUSCA. Hela harpularia H. and A. Adams (Couthouy). B. pleuroto)7iaria H. and A. Adams (Couthouy). JB. turricida (Montagu). Buccimim utidatum Linne. Neptunea curta Verrill (Jeffreys sp., Fusus Islatulicus Gould). N. pygmoea H. and A. Adams (Gould). Lunatia heros H. and A. Adams, variety triserlata (Say). L. immaculata H. and A. Adams (Totten). Turritella erosa Couthouy, T. acicula Stimpson. Margarita obscura Gould (Couthouy) ; common. Diodora noachina Gray (Linne). Hauleia mendicaria Cai-penter ; large specimens. Mactra so^^V?^ss^m a Chemnitz (Gray). Gyprina Islandiea Lamarck (Linne). Gardium pinnulatnm Conrad ; common. Astarte quadrans Gould. Leda tenuisidcata Stimpson (Couthouy) ; common. Grenella glandida H. and A. Adams (Totten). Modiola modiolus Turton (Linne). Modiolaria Imvigata (Gi-ay). Pecten tenuicostatus Mighels. P. Islandicus Chemnitz (Mtiller). Anomia aculeata Gmelin. Glandula arenicola Verrill. Gellularla ternata Johnston (Busk). Gemellaria loricata Busk (Linne). Bugula Murray ana Busk (Bean). Cellepora tuberosa D'Orhigny. Smith and Harger — St. George's Banks Dredgings. 21 Kadiata. Lophothuria Fabricii Verrill, Strongylocentrotus Drbhachiensis A. Agassiz ; common. Ecliinarachnius parma Gray ; very abundant. Solaster endeca Forbes. Crossaster papposus Mtiller and Troschel. Crihrella sanguhiolenta Liitken. Asterias mdgaris Stimpson. Leptasterias Stimpsoni Verrill. OphiophoUs acideata Gray ; common. Ophioglypha robusta Lyman. Eudendriuni rarnosum Ehrenberg (Linne). E. capillare Alder. Campanularia verticillata Lamarck ; common. C. Hincksii Alder. Lafoea dumosa Sars (Fleming) ; common. Calycella syringa Hincks (Linne). Grammaria ahictina Sars {G. robusta Stimpson). Coppinia arcta Hincks ; on hydroid stems. Halecium labrosum Alder. Sertularella polyzonias Gray (Linne) ; common. " " variety gigantea Hincks. S. tricvspidota Hincks (Alder) ; abundant. Sertularia abietina Linne; one, very large. S. latiuscula Stimpson. S. cupressina Linne ; common. S. argentea Ellis and Solander. [Plate HI, figure 2.] Hydvallmania fidcata Hincks; abundant. Third (r), north and a little east of the last, latitude 42° 3', longi- tude 67° 31', in 40 fathoms, coarse sandy bottom: Crustacea. Eupagnrus Bemhardns Brandt. Annelida. Dodecaceria concharum CErsted. Spirorbis quadrangidaris Stimpson. MOLLUSCA. Bela harpidaria H. and A. Adams (Couthouy). Natica clausa Broderip and Sowei-by. Scalaria Groenlandica Sowerby. Margarita obsciira Gould (Couthouy). Mactra solidissima Chemnitz (Gray). 22 Smith and Harger — 8t. George's JBanhs Dredgings. Cyprina Islandica Lamarck (Ijinne). Cardiwrn pinnulatum Conrad. Astarte castanea Say. Pecten temdcostatus Migliels. Cellularia sp. Bugula Murrayana Busk (Bean). Radiata. Echinarachnms parma Gray. Hydractinia polyclina Agassiz. Sertularella polyzonias Gray, var. gigantea Hincks. Sponges. Ghalina oculata Bowerbank, and a massive siliceous sponge. Fifth {s) a little to northeast of the bank, latitude 42° 11', longi- tude 67° iV, two hauls in 150 fathoms, soft sandy mud with a few pebbles : Crustacea. Hyas coarctatus Leach. Eupagurus Bernhardus Brandt. E. Kroyeri Stimpson. E. puhescens Brandt. Ptilocheirus pingxiis Stimpson. JEga psora Bate and Westwood. Conilera polita Harger (Stimpson). JBalanus porcatus Costa. Annelida. LcBtraonice filicornis Kinberg. Harmothoe imbricata Malmgren (Linne). Antinoe angusta Verrill, sp. nov. Eucranta villosa Malmgren. Nepthys ingens Stimpson. Eumbrieonereis /ragllis CErsted (Miiller). [Plate V, figure 2.] Nothria conchylega Malmgren (Sars). [Plate VII, figure 3.] N', opalina Verrill. [Plate VII, figure 4.] Goniada maculata Ql^i'sted. Scalihregma inflation Rathke. Spiochcetopterus (?) ; tubes exactly like those of this genus. Scolecolepis cirrata Malmgren (Sars). Ancistria capillaris Verrill. Maldane Sarsii Malmgren ; abundant. Smith and Sarger — St. George's Banks Dredging s. 23 Cistenides granulata Malmgren. Amphicteis Gunneri Malmgren (Sars). Amage auricula Malmgren. Smythella elongata Verrill. Melinna cristata Malmgren (Sars). Pista cristata Malmgren (Sars). [Plate IV, figure 2.] Grymcea spiralis Verrill. [Plate IV, figure 1.] Terebellides Stroemi Sars. Protula borealis Sars ? ; tubes only. Gephyrea. PhasGolosoma cmmentariutn Verrill. P. boreale Keferstein (?) P. tubicola Verrill. MOLLUSCA. Adniete viridula Stimpson (Miiller). Neptunea pygmoia H. and A. Adams (Gould) ; abundant. Ringicida nitida Verrill ; one living. [Plate I, figure 2.] Lunatia Grmnlandica (M oiler). Natica clausa Broderip and Sowerby. Veliitina zonata Gould. Torrellia vestita Jeffreys ; one specimen. Aporrhais occidentalis Sowerby. Margarita cirierea Gould. M. obscura Gould (Couthouy). Trachydermon album Carpenter. Scaphander puncto-striatus Stimpson ; one very large. Gylichna alha Loven (Brown). Entalis striolata Stimpson ; abundant. [Plate I, figure ;^.] Dentalium occidentale Stim})son ; one specimen. Nemra arctica Sars Thracia myopsis Beck. Cardium pinmdatum Conrad. Astarte quadrans Gould. A. lens Stimpson, dwarf variety ; common. Yoldia obesa Stimpson. Deda tenuisulcata Stimpson (Couthouy). Area peetunculoides Scacchi ; several. Pecten pustulosus Verrill ; one living. Anomia aculeata Gmelin. Glandula arenicokt Verrill ; common. Terebratulina septentriorialis (Couthouy) ; abundant. 24 Smith and Harger — St. George's Banks Dredgings. Hornerea lichenoides Smitt (Linne). Discoporella verrucaria Smitt, Gellularia ternata Johnston, var. gracilis {arctica Busk sp.). G. Peachii Busk. Gcd)erea Ellisii Smitt (F'leming). JBugula Murray ana Busk (Bean) ; abundant. jB. avicularia Busk, var. fastigiata. Gellepora scahra Smitt (Fabricius). G. ramulosa (Linne). Radiata. Thyone scahra Verrill. Schizaster fragilis Agassiz (Duben and Koren) ; several. Echinarachnius parina "Gray. Solaster furcifer Duben and Koren ; one specimen. Archaster arcticus Sars ; one specimen. Ophioglypha Sarsii Lyman. O. affinis Lyman. Ophiacantha spinxdosa Milller and Troschel. Glytia Johnstoni Hincks. Eudendrium capillar e Alder. Sertularella tricuspidata Hincks (Alder). S. Gayi Hincks (Lamoroux). Sertularia cnpressina Linne. Pennatula aculeata Danielsen. Virgidaria Lyungmanii KoUiker. Bolocera Tuedice Gosse ; tentacles only. Urticina crassicornis Ebrenberg. GeHanthus borealis Verrill. [Plate II, figure 5.] The lists of species from all the localities (a, 5, c, d, e, /', q, r,) on George's Bank itself, show that tlie fauna of that region is almost exactly the same as in the Bay of Fundy, at the same de))ths and on similar bottom. To be sure, on the one hand, several arctic species, not yet found in the Bay of Fundy, occurred upon the Bank ; but on the other hand, several apparently more southern forms were found, as the species of Grepidxda and Stylifer. The two dredgings {h and i) upon LeHave Bank seem to indicate, as we might expect, a somewhat more arctic fauna than that upon George's Bank, since several arctic species, not known from George's Bank or the Bay of Fundy, occurred there, though Grepidula phiita was also found. The dredgings in deep water near the Banks indicate a fauna quite different from that upon the Banks themselves. This is undoubtedly Smith and Uarger — St. George's Banhs Dredgings. 25 partially owing to the diiference in the character of the bottom as well as to the diiference in depth. Of the species occurring in deep water, a much larger proportion than in the shallower waters are the same as those of northern Europe, At the greatest depth reached by the dredge, about 430 fathoms, at the locality [g) east of George's Bank, almost all the species which were not also found in shallow water are European. Some of these species, however, were dredged in 1872, by Prof. Veri-ill, in the central part of the Bay of Fundy, east of Grand Menan Island, in 95 to 106 fathoms, where the char- acter of the bottom was quite similar to that of our deepest dredg- ing. At each of the three deepest of Dr. Packard's dredgings, (o) 110 fathoms, {p) 85 fathoms, and (s) 150 fathoms, the bottom was com- posed of soft sandy mud, very ditferent in character from that at any of the localities examined by us. The fauna of the bottom at these three places was essentially the same, and, although many of the species, on account of the character of the bottom, were diiferent from those at the locality in 430 fatlioms, about the same proportion are identical with European species. Although the dredgings in deep water were so few, the facts pre- sented in the foregoing lists with reference to the bathymetrical dis- tribution of species, are important and very interesting. Of the species enumerated from 430 fathoms, considerably more than half are well known shallow water forms, many of them even occurring between tides in the Bay of Fundy and at other points on the coast, while nearly all the species mentioned are also found at less than 50 fathoms depth. The same remarks Avill apply to the deeper dredg- ings of Dr. Packard and Mr. Cooke. The species from the deepest dredging belong apparently to as highly organized groups of animals as do those from shallow water. We were not able to detect any decrease in the intensity of the colors in individuals from this depth. The colors of Pandalus an?iidlcor>ns, Eupagurus Kroyerl^ Unciola in-orata^ and Urticina crassicornis, all brightly - colored species, seemed to have lost none of their intensity at the depth of 430 fathoms. Besides the investigation of the fauna of the bottom by means of the dredge, every opportunity Avas employed for collecting those animals which live in })art or wholly at the surface of the water. Notwithstanding the unfavorable character of the weather during most of the time we were at sea, towing nets were used, whenever soundings were being made, and usually with very good results. Nets of small size were several times successfully used even Avhen the Trans. Conn. Acad., Vol. III. 4 August, 1874. 26 Smith and Harger — St. George's Banks Dredglngs. steamer was under weigh. In this way a great number of surface species were collected, and a large proportion of them are additions to the fauna of our coast. Many of these species belong to genera previously known only from much farther south, or from the eastern or southern Atlantic, while quite a number are undescribed. August 29, on and near Cultivator Shoal {k), where the surface temperature of the water was 62°, the following were taken : Trachy- nema digitale A. Agassiz, Pleurobrachia rhododactyla Agassiz, species of Sagitta and Aidolytus, several species of Copeopod Crustacea, Calliopius Icevhisculus Boeck (among floating rock-weed), the young of some Brachyuran in the zoea and megalops stages of growth, and a species of 31otella (?). East of George's Bank, in latitude 41° 20' to 30', longitude 63° to 63° 30', September 14, during the day, many species were taken, but as they all occurred, with many additional species, on the follow- ing day, it is not necessary to enumerate them separately. On the evening of September 14, from nine to ten o'clock, still east of the Bank (m), in latitude 41° 25', longitude 63° 55', while the sur- face temperature was 65°, the following forms occurred : Pleuro- hrachia sp. ; a species of Salpa in abundance ; several species of Heteroi^ods and Pteropods, among the latter Sjnrialis Gouldii Stimpson, and species of Styliola j a species of Sagitta,' a species of Sapphiri^ia and a great many other Copeopods ; species of Syj^eria, Phrosina^ and of another allied genus ; a species of Thysa)iopoda, which was beautifully phosphorescent ; young Brachyura in the zoea and megalops stages, and the young of some Macrouran. September 15, on the same line of soundings, in latitude 41° 25', longitude 65° 5' to 30', the temj^erature of the water varying from 66° to V0°, but most of the time at the latter point, very many species occurred, and among them the following: Physalia pelagica Lamarck (Portuguese man-of-war), Cestum Veneris Lesueur (?), Stomolophus meleagris Agassiz, Charyhdea periphylla Peron and Lesueur, Pelagia cyanella Peron and Lesueur ; species of Salpa and Sagitta in great abundance ; Lepas pectinata Spengler and L. fascicularis Ellis and Solander ; two species of Sapphirina and many other genera of Coj^e- opoda ; species of Oxycephalus, Platyscelus, Pronoe^ Anchyloinera^ ThyropuSy Phronima (?), and Hyperia; Calliopius Imviiiscidus Boeck, common among floating rock-weed ; species of Lucifer and Mysis ; Latreiites ensiferus Stimpson, JVautdognq)sus mimdus Edwards, and Neptunus Sayi Stimpson among gulf-weed, and the latter frequently seen swimming at some distance from the sea-weed ; three species of Heteropods and ten species of Pteropods, all new to our coast. Smith and Harger — St. George's Banks Dredgings. 27 Among the Pteropods are Styliola acus (Eschscholtz sp.), and four other species of the same genus, two of Pleuropus, Spirialis Gouldii, etc. Many of these species and genera are quite new to the fauna of the United States, and nearly all of them to the coast of New England. They are nearly all, as far as known, characteristic Gulf Stream forms. Notes on some of the Species enumerated ; by S. I. Smith.* Crustacea. Eupagurus Bernhardus Brandt. Pagurus Bernhardus (Linne sp.) Fabricius, Entomologia systematica, ii, p. 469, 1793, and Supplementum, p. 411, 1798. Pagurus (subgenus Eupagurus, section SirejJtodactylus) Bernhardus Brandt, Midden- dorff's Sibirische Reise, Krebse, p. 106, 1851. Eupagurus Bernhardus Stimpson, Crust. Pacific Shores of North America, Journal Boston Soc. Nat. Hist., vi, p. 483 (separate copies, p. 43), 1857. 1 have recently f wrongly given Stimpson as authority for this and the next species, not having at the time access to Brandt's work, and not being able to comprehend his absurdly complex nomenclature from the quotation of his names by other authors. Eupagurus pubescens Brandt. Pagurus p>ubescei-ts Kroyer (in part), Gronlands Amfipoder, p. 68, 1838, and Natur- historisk Tidsskrift, ii, p. 251, 1839. Pagurus (subgenus Eupagurus, section Orthodactylus) pubescens Brandt, op. cit., p. Ill, 1851. Eupagurus pubescens Stimpson, Prodromus descriptionis Animalium evertebratorum, etc.. Proceedings Acad. Nat. Sci., Pliiladelphia, 1858, p. 237 (separate copies, p. 75), 1859, and Notes on North American Crustacea, Annals Lyceum Nat. Hist., New York, vii, p. 89 (separate copies, p. 43), 1859. This species is common on our eastern coast north of Cape Cod, but is not quite as abundant as the last species and is seldom found at low water. South of Cape Cod it is apparently confined to the deeper and colder waters. *With the exception of the portion relating to the Crustacea, these notes have had the benefit of Professor Verrill's revision, and the descriptions of all the new species have been copied from his pubhshed papers, or, in the case of those here for the first time described, have been prepared by him specially for these pages, and are marked by his initials. \ Report upon the Invertebrate Animals of Vineyard Sound, in Report of the U. S. Commissioner of Fish and Fisheries, Part I, 1873 (published in 1874). 28 Smith and Harger — St. George's Banks Dredgings. Eupagurus Kroyeri Stimpson. Notes on North American Crustacea, Annals Lyceum Nat. Hist., vii, p. 89 (43)' 1859. This species is A^ery closely allied to the last and is very easily con- founded with it, especially when young. The diflerences in the rela- tive proportions of the chelipeds and ambulatory legs, given by Stimpson, will not hold for distinguishing the two species, but the diiferences in the amount of pubescence and especially in the form and armature of the chelipeds seem to be constant characters, suffi- cient for distinguishing them. The Kroyeri has about the same range, on our coast, as the last species, although I have never seen it south of Cape Cod, but is apparently less abundant and more confined to the deeper waters. Sabinea septemcarinata Owen (Sabine sp.) This species was dredged in 68 fathoms off Casco Bay in the sum- mer of 1878. It has also been found by Mr. Whiteaves in the Gulf of St. Lawrence and by Dr. Packard on the coast of Labrador. It is an exceedingly arctic and circumpolar species. Caridon Gordoni Goes (Bate sp. ?) Goes, Crustacea decapoda podophthalma marina Suecipe (from (Efversight af Kongl. Vetenskaps-Akad. Forliandlingar, Stockholm, IHtiS), p. 10. We have dredged this species in 50 fathoms in the Bay of Fundy, and Dr. Packard and Mr. Cooke obtained it on Cashe's Ledge in 1873. Our specimens agree well with the detailed description given by Goes, except that they have a well developed epipodus (" flagellum") upon the second, third and fourth cephalothoracic legs, as in some species of Hippolyte^ while Goes says of the second legs, " nee palpo nee (quoad viderim) flagello ullo instructis," and of the third to fifth, " flagellum basale nullum inspicere potui." From the guarded manner in which Goes mentions these wholly negative characters, I am inclined to re- gard them as doubtful. Our specimens agree so completely in all other respects that it seems highly improbable that they should be distinct from the European species. Diastylis quadrispinosa G. 0. Sars. (Efversight af Kongl. Vetenskaps-Akademiens Forhandlingar, 1871, Stockholm, p. 27 ; and Beskrivelse af de Paa Fregatten Josephiens Expedition Fundne Cumaeeer, in Kongl. Svenska Vetenskaps-Akademiens Handliugar, ix, p. 28, plates 10, 11, figs. 51-61, 1871. This is the most abundant species of the genus from off Buzzard's Bay and Vineyard Sound to Nova Scotia. It ranges north at least as far as the Gulf of St. Lawrence. Smith and Harger — St. George's Banks Dredgings. 29 Diastylis sculpta G- 0. Sars. Loc. cit., CEfversight, p. 71 ; Handlingar, p. 24, pis. 1-9, figs. 1-49. This species is not uncommon in Casco Bay and the Bay of Fundy. Phoxus Kroyeri Stimpson. Marine Invertebrates of Grand Manan, p. 58, 1853. We have dredged this species in 10 to 29 fathoms in and off Viue- yai'd Sound, on sandy and muddy bottoms in shallow water in Casco Bay, and have found it from low water to 20 fathoms in the Bay of Fundy. Mr. Whiteaves has dredged it in the Gulf of St. Lawrence in 200 fathoms, muddy bottom. Our species is very closely allied to, and probably identical with, the P. Holbollii Kroyer which is found in Greenland, Iceland and northern Scandinavia. Harpina fusiformis Smith. Phoxus fusiformis Stimpson, Marine Invertebrates of Grand Manan, p. 57, 1853. This species is very likely identical with the II. plwiiosa Boeck {Phoxus plumosus Kroyer), which has very nearly the same range as Phoxus Holhollii. We have dredged our species in 20 to 60 fathoms, muddy bottom, in the Bay of Fundy. Mr. Whiteaves has dredged it frequently, in the Gulf of St, Lawrence. Stenothoe peltata Smith, sp. nov. Plate IV, figures 5 to 8. Female. Eyes round and nearly white in alcoholic specimens. Antennuloe considerably shorter than the epimera of the fourth seg- ment; first segment of the peduncle stout, fully as long as the head, the second shorter, and the third very short and like the segments of the flagellum ; flagellum scarcely longer than the peduncle, com- posed of about eight segments. Antennje slightly longer than the antennula?; the ultimate and penultimate segments of the peduncle about equal in length; flagellum about as long as the flagellum of the antennulae. Second epimeron (figure 5) rudely ovate, twice as high as broad ; third somewhat rectangular, no wider than the second but considerably deeper ; foiirth (figure 6) very large, slightly deeper than the third and a third or a fourth longer than deep, being about as long as the first five segments of the thorax, the inferior margin regularly curved and the posterior convex in outline. First legs (figure 7) small and slender; merus triangular and broader distally than the carpus, which is not quite twice as long as broad and has the lateral margins parallel ; propodus narrower but slightly longer than the carpus and narrowed distally ; dactylus about half as long as the jjro- 30 Smith and Harger — St. George's Banks Dredgings. podus. Second legs (figure 5) stouter; merus short triangular; carpus much broader than long and only slightly produced beneath the propodus; propodus about as long as the breadth of the epinieron, nearly twice as long as broad ; palmary margin (figure 8) convex in outline, slightly oblique, with an acute lobe and a spine at the pos- terior angle, within which the tip of the dactylus closes. Third and fourth legs slender and nearly naked. Basal segment in the fifth legs slender, foitr times as long as broad, not wider than the merus. Sixth and seventh legs slightly shorter than the fifth, the basal segments posteriorly dilated and squamiform in both pairs, but broader in the seventh than in the sixth. Posterior caudal stylets with the ramus slightly longer than the peduncle. Length of largest specimen, from front of head to tip of telson, about 6™"'. The mandibles are without palpi or molar tubercles, and in all other characters the species agrees with the genus Stenothoe as restricted by Boeck, but it seems to be very distinct from either of the European species. Near Cultivator Shoal (haul Z*), 30 fathoms, soft, sandy bottom, August 29. Syrrhoe crenulata Goes. Crustacea amphipoda maris Spetsbergiam alluentis, CEfversight af Kongl. Vetens- kaps-Akad. Forhandlingar, Stockholm, 1865, p. 527. pi. xl, fig. 25; Boeck, Crus- tacea amphipoda borealia et arctica (Vidensk.-Selskabs Forhandlinger, Christiania, 1870), p. 67, 1870. We have also dredged this species, in 1872, in 12 fathoms in John- son's Bay, near Eastport, Maine, and in 90 to 100 fathoms off Grand Menan, and have examined specimens dredged, in 1873, in 30 fath- oms, in Gaspe Bay, Gulf of St. Lawrence. Our specimens have all been considerably larger than the one figured by Goes, but otherwise agree perfectly. It seems to be an exceedingly arctic form, being found in Europe from Spitzbergen to the western coast of Norway. Tiron acanthurus Liiijeborg. Boeck, op. cit., p. 69. Syrrhoe hicmpis Goes, loc. cit., p. 528, pi. xl, fig. 26. f Thes- sarops hastata Norman, Annals and Magazine Nat. Hist., IV, ii, p. 412, pi. xxii, figs. 4-7, 1868. This species has apparently not been noticed on our coast before. It has been found in Greenland, Finmark, and on the western coast of Norway, while Norman's TJcessarops was from the English coast. CEdiceros lynceus Sars. Oversigt over nordsk-arct. Krebsdyr. Forhandl. i Vidensk-Selsk. i Christiania, 1858, p. 143 (teste Boeck); Boeck, op. cit., p. 82. CEdiceros propinquus Goes, loc. cit.. Smith and Harger — 8t. George's Banks Bredgings. 31 p. 526, 1865, pi. xxxix, fig. 19. Monoculodes nuhilius Packard, Memoirs Boston See. Nat. Hist, i, p. 398, 1867. We dredged this species in the Bay of Fundy in 1868 and 18Y2, the latter year in 60 to 80 fathoms ; in Casco Bay, in 27 ftithonis, in 1873, and Dr. Packard and Mr. C-ooke obtained it at several local- ities, in the " Gulf of Maine," from 50 to 90 fathoms, on the expedi- tion of the Bache in 1873. I have also examined specimens dredged in the Gulf of St. Lawrence by Mr. Whiteaves and on the coast of Labrador by Dr. Packard. It extends to Greenland, Iceland, Spitz- bergen and Finmark. Monoculodes borealis Boeek. Op. cit, p. 88, 1870. (Ediceros affinis Goes, loc. cit., p. 527, pi. xxxix, fig. 21', 1865 (non Bruzelius). This species is recorded from Spitzbergen and northern Norway by Goes and Boeck, but seems not to have been noticed on this side of the Atlantic before. Paramphithoe pulchella Bruzelius (Kroyer sp.) We have dredged this species off Casco Bay and in the Bay of Fundy, on hard bottoms, in from 40 to 90 fathoms, and it was dredged on Cashe's Ledge and Stellwagen's Bank, in 1873, by Dr. Packard and Mr. Cooke. It extends north to the Gulf of St. Law- rence, and, according to Boeck, to Greenland, Iceland, Spitzbergen, and the western coast of Norway. Paramphithoe cataphracta Smith. AmpMthonotus cataphi-adus Stimpson, Synopsis of the Marine Invertebrata of Grand Manan, p. 52, 185.'i (description copied in Bate, Catalogue of Amphipodus Crus- tacea in the British Museum, p. 152, 1862.) This species is apparently a true Paramphithoe, as restricted by Boeck, and closely allied to, if not identical Avith, P. pano2)la Bru- zelius [Aynphithoe pjaaopla Kroyer). Boeck places Pleustes tiibercii- latus Bate as a synonym of Kroyer's species, and if he is correct in this, our species is undoubtedly distinct. The cataphracta appears to be an inhabitant of hard or coarse sandy and shelly bottoms from 5 to 50 fathoms. We have dredged it sparingly in Casco Bay and the Bay of Fundy, and Dr. Packard has dredged it on the coast of Labrador. VertUmnuS serratUS ? Goes (Fabricius sp.) Acanfhonotm serratus Stimpson, Synopsis of the Marine Invertebrata of Grand Manan, p. 52, 1853. Our specimens all differ from the descriptions and figures given by Boeck and Kroyer in the armature of the posterior margin of the 32 Smith and Harger — St. G-eorge's Banks Dredgings. thii'd segment of the abdomen. In our specimens the upper process from this margin is armed with four or five teeth above and at the tip, while the lower process is armed with five or six teeth similarly situated, but with no teeth on the lower margin except just at the tip. In Kroyer's figure (Gronlands Ampfipoder, plate ii, figure 8) the upper process is represented as terminating in a single tooth and the lower process as toothed along both sides ; Boeck's description agrees with this except that he says there are two teeth at the tip of the upper process. It is not uncommon on hard bottoms in from 5 to 50 fathoms in the Bay of Fundy. We have also dredged it in Casco Bay and have received it from the Gulf of St. Lawrence, where it was dredged by Mr. Whiteaves. Acanthozone cuspidata Boeck. This species is quite common on hard, and especially on spongy bottoms in 5 to 40 fathoms in the Bay of Fundy, although it is not mentioned by Stimpson in his work on Grand Menan. We have also dredged it in Casco Bay, and Mr. Whiteaves has obtained it in the Gulf of St. Lawrence. It ranges to Greenland, Spitzbergen and Finmark. BybliS Gaimardi Boeck (Kroyer Bp.) We have frequently dredged this species in Casco Bay and the Bay of Fundy, on muddy bottoms in 10 to 60 fathoms. It extends north to the Gulf of St. Lawrence (Whiteaves), Labrador (Packard), and, according to Boeck, to Greenland, Iceland, Spitzbergen and Norway. The Ampelisca Gahnardi of Bate, and Bate and West- wood, is not this species but a true Ampelisca. All the species of this sub-family are undoubtedly tube dwellers. Lilljeborg noticed the habit in HaploOps ; it has been observed in species of Ampelisca by Professor Verrill and myself. In this species, the glands which secrete the cementing fluid are situated principally in the meral and basal segments of the third and fourth pairs of thoracic legs. Xenoclea megachir Smith, sp. nov. Plate IV, figures 1 to 4. Male. Eyes large, black, very slightly elongated, and approaching closely the edges of the triangular prominence of the inferior angle of the front margin of the head. Peduncle of the anteninila' about as long as the head and the first two segments of the thorax, the second segment longest, the first and third about equal in length, Smith and Sarger — St. George's BanJcs Dredgings. 33 flagellum about as long as the peduncle and composed of twelve to sixteen segments. Antennae a little longer than the antennulae ; ultimate and penultimate segments of the peduncle sub-equal in length ; flagellum slightly shorter than the peduncle and composed of eleven to fifteen segments. First epimeron (figure 1) as broad as high ; second (figiu-e 2) broader than high ; third (figure 3) and fourth not broader than high and successively deeper than the first and second ; fifth (figure 4) slightly deeper than the fourth and its terminal portion as broad. In the first legs (figure 1), the carpus longer and broader than the propodus, which is somewhat oval and twice as long as broad ; the dactylus slender, slightly curved and fully as long as the propodus. The inferior distal margin of the pro- podus is regularly curved to a short distance from the extremity, where there is a small but deep emargination, beyond which and round upon the short distal margin the edge is serrate with minutely crenulated teeth ; the posterior margin is furnished with numerous slender seta? and with a single stout spine at the emargination near the distal end. The inner edge of the dactylus is armed with a series of acute teeth directed obliquely toward the tip. In the second pair of legs (figure 2) the propodus is very stout, about twice as long as the epimeron and scarcely one-half longer than broad ; the palmary margin oblique and armed near the middle with two stout obtuse teeth ; the dactylus stout and its inner edge sinuous. Third (figure 3) and fourth pair of legs alike ; ischium and carpus short, each nearly or quite as broad as long; merus fully as long as the epimeron and half as broad as long ; propodus slender, not more than half as broad as the carpus but twice as long ; dactylus slender, about half as long as the propodus. Basal segment in the fifth legs (figure 4) squamiform, oval, nearly as broad as long and with a mai-ked angular emargination at the inferior posterior angle ; carpus only slightly longer than the breadth of the raerus ; dactylus slightly curved and acute. Second and third segments of the abdomen with the inferior portion of the posterior margin sinuous, and the inferior angle prominent, but scarcely less than right-angled. The outer rami in all the caudal stylets slightly shorter than the inner, and all the rami armed with short spines above and more slender spines at the tips. Telson stout, about as broad as long and scarcely more than half as long as the peduncle of the posterior caudal stylets, the posterior margin with a few setiform hairs each side. In the female the hands in the second pair of limbs are propor- tionally much smaller and more abundantly provided with hairs, while the teeth, or lobes of tha palmary margin, are further apart and Teans. Conn. Acad., Yol. III. 5 August, 1874. 34 Smith and Harget — St. George's Banks Dredgings. separated by a broad and deep, rounded sinus ; the dactyhis is not so stout, and has the inner margin evenly curved and serrated. Length, from front of head to tip of telson, 5 5 to Y'o""'. I refer this species with some hesitation to Boeck's genus Xenoclea, which is known to me only from the very short diagnosis of the genus and of the single species X. Batei, given in his Crustacea Amphipoda Borealia et Arctica, p. 155. " Pedes 3tii et 4ti paris articulo Imo latissimo" of the generic diagnosis would scarcely apply to our species, but in all the other generic characters it agrees perfectly, as it does also with the diagnosis of the sub-family Photinoe, except that the mandibles each bear six serrated spines instead of the usual numbei-, four. Near Cultivator Shoal (haul J), 30 fathoms, soft, sandy bottom, August 29 ; and on the northern side of George's Bank (haul q), north latitude 42°, west longitude 67° 42', 45 fathoms, coarse sandy bottom. Also, in 18 fathoms, off Watch Hill, Rhode Island. When first examining the alcoholic specimens of this species, I noticed a peculiar opaque glandular structure filling a large portion of the third and fourth pairs of thoracic legs, which in most, if not all, the non-tul)e-building Amjihipoda are wholly occui)ied by muscles. A further examination shows that the terminal segment (dactylus) in these legs is not acute and claw-like, biit truncated at the tip and apparently tubular. In this sjiecies, a large cylindrical portion of the gland lies along each side of the long basal segment, and these two portions uniting at the distal end pass through tlie ischial and along the jiosterior side of the meral and carpal segments and doubt- less connect with the tubular dactylus. (See Plate III, figure 3.) There can be no doubt that these are the glands which secrete the cement with which the tubes are built, and that these two pairs of legs are specialized for that purpose. A hasty examination revealed a similar structiu'e of the corresponding legs in Ainjy/iithoe macidata, Ptilocheirus pinguis, Cerapus rubricomis, Byblis Gaimardi, and a species of Ampelisca. In all these except the last two a very large proportion of the gland is in the basal segment. In the Amphithoe this segment is thickened and the gland is in the middle. In the Cerapus it is very broad and almost entirely filled by the gland, with only very slender muscles through the middle, and the orifice in the dactylus is not at the very tip but sub-terminal on the posterior side. In the Ptilocheirus the gland forms three longitudinal masses in the basal segment and is also largely developed in the meral and carpal segments. The dactylus is long and slender and the orifice sub-ter- minal. In Ampelisca and Byhlis (which, like Saploops, are tube- building genera) the meral segments of the specialized legs are nearly Smith and Harcjer — St. George's Banks Dredgings. 35 as large as the basal and contain a proportionally large part of the gland. ScapellTim Stroemi Sars. Plate in, fignre 9. I am not aware that a description of this species has yet been pub- lished, although the name was used by Prof. Michael Sars in his list of animals living at great depths in the sea, published in 1869,* and the species has since been incidentally figured, without any detail, on the stems of Mopsea borealis, by Dr. G. O. Sars in his recent work on " Some Remarkable Forms of Animal Life from Great Depths off" the Norwegian Coast" (Plate V, figure 2). Dr. G. O, Sars has, how- ever, very kindly compared a drawing of one of our specimens, and he writes me that it agrees in every detail Avith the Norwegian form. It is very distinct from any of the species described in Darwin's great work, and also from the species recently described from the Challenger Expedition. Since our specimens were obtained from 430 fathoms. Dr. Packard and Mr. Cooke have dredged in 50 to 70 fathoms near Cashe's Ledge, and in 142 fathoms, 20 miles east of Cape Race (both localities within the " Gulf of Maine"). AH the specimens were attached to stems of hydroids. On the Norwegian coast the species has the same habit and has been found by Dr. G. O. Sars in from 80 to 300 fathoms. Annelida. Lsenilla (?) mollis 6. 0. Sars. Bidrag til Kundskaben om Christianiafjordens Fauna, iii, p. 7, plate xiv, figs. 1-12, 1873. Body large, rather stout, medially convex. Head short and broad, narrowed posteriorly, prominently rounded laterally, and pro- duced into two very small conical points anteriorly. The anterior eyes are larger than the others, situated on the outer and upper sur- face of the lateral prominences, and look outward and upward ; the posterior pair are nearer together, on the lateral slopes of the nar- rowed part of the head. The median tentacle is wanting in our speci- men, but its basal segment is of moderate size and cylindro-conical ; the antennje are slender, and nearly three times the length of the head, banded with brown ; the palpi are rather slender and regularly tapered, smooth, or nearly so, four or five times the length of the head. The dorsal and tentacular cirri and the scales are wanting in the single specimen obtained. The lateral appendages are large and * Forhandlinger i Videnskabs-Selskabet i Christiania, 1868, p. 259, 1869. 36 Smith and Harger — St. George's J^cmJcs Dredgings. prominent, with large fascicles of long, slender seta? in the lower rami, and much shorter and stouter ones in the upper rami. The appenda- ges, including setae, equal or exceed the breadth of the body. Breadth of body, exclusive of appendages, 7™™ ; length of the latter, without setaa, S-S"^"" ; with setaj, 10'""; length of body to the 15th segment, 25™"\ The setae of the upper ramus are very stout, and all of nearly the same form, the upper ones being merely smaller and stouter than the rest ; they are nearly straight or slightly recurved, with rather conspicuous, moderately close transverse series of denticles, which ex- tend nearly to the ends, leaving only stout, naked, straight tips. The setae of the lower ramus are much longer and far more slender, with a long, slender shaft, and a slightly expanded terminal portion, which is conspicuously, but not closely, spiniilated on both sides to the tips ; many of these are nearly straight, but most are slightly curved ; the upper ones are most slender, and mostly have the tips only very slightly bidentate, and the spinules exceed the diameter of the setae and increase toward the end, the last ones projecting considerably beyond the tip ; the middle ones are about twice as stout, having the terminal part more expanded ; their spinulation is similar, but the tips are more distinctly, though slightly, bidentate, the denticles be- ing partially obscured by the terminal spinules that project beyond them ; the lower ones are moi-e slender and like the upper ones in form and character. Near St. George's Bank, 110 fathoms, mud. Coast of Norway, 40- 200 fathoms (G. O. Sars). Our specimen is imperfect, but the head and seta? are quite peculiar. The latter are remarkable for the length of the spinules, and for the minuteness of the denticles at the tips. — A. E, V. Antinoe angusta Verriii, sp. nov. Body narrow, rather slender, elongated, tapering gradually pos- teriorly. Head small, short, rounded, broader than long, the lateral lobes short, not prolonged into points anteriorly, but obtusely rounded ; the lateral borders also well rounded. Eyes small, nearly equal ; the postei-ior pair situated on the dorsal side of the vertex ; the anterior pair farther apart on the outer and upper surfoce of the lateral promi- nences. Tentacle long and very slender, about three times the length of the head; antennae small and short, scarcely one-third as long as the head ; palpi moderately large, glabrous, considerably longer than the tentacle. Dorsal cirri slender, pretty regularly but not closely covered with slender papillae. The lateral appendages, except ante- riorly, bear large fascicles of long, fine capillary setae, which gives a Smith and Harger — 8t. George's Banks Dredgings. Si villous appearance to the sides. The elytra, in our specimen, are wanting. The color, in alcohol, is light brown, crossed by lighter transverse lines. Length, 15™'" ; breadth, without appendages, 2""" ; breadth, including setae, 4'"™. On the middle segments the setae of the upper ramus are quite unequal in size and length ; the upper ones are stout, with the ends more or less recurved ; the middle ones are still larger and more than twice as long, slightly curved, and, like the former, conspicuously transversely serrulate almost to the extreme tips ; the lower ones are shorter, less stout, and slightly curved. The seta3 of the lower ramus are longer and extremely slender; the upper ones are mostly but slightly expanded in the middle, with very long, flexil)le capillary tips, finely tapered to the end, and very minutely serrulate or nearly smooth ; the median ones are stouter, more expanded in the middle, with long, acuminate, slender, sharp tips, and with conspicuous, rather distant spinules on one or both sides, which become very fine and moi'e crowded distally ; tlie lower ones are much shorter, and have shorter but still very slender tips, and fewer and more distant spi- nules. The ventral cirri are slender, tapered, with few, distantly scat- tered, small papillae. — A. E. V. Near Saint George's Bank, 150 fathoms, mud (locality s). Antinoe Sarsi Kinberg. Maliagren, Nordiska Hafs-Annulater, (Efversigt Kongl. Yetenskaps-Akad. For- handlingar, Stockholm, 1865, p. 75, pi. 9, fig. G ; Annulata PolychiBta, p. 13, 1867. Our specimen of this species agrees very well with jMalmgren's figures and description. It is much larger and stouter than the pre- ceding, and the head is longer and quite different in form, the lateral lobes extending forward into acute conical points. The set£e are similar to tliose of the former, but the median and in- ferior setae of the lower ramus are relatively somewhat stouter and have the tips less attenuated and elongated, while the spinules are larger and more conspicuous, especially on the upper setae of the lower ramus. Near Saint George's Bank, 85 fathoms, mud. Gulf of Saint Law- rence (Whiteaves, t. Mcintosh). — A. E. V. Encranta villosa Maimgren. Eucranta villosa Maimgren, Nordiska Hafs-Annulater, CEfversigt af Kongl. Vetens- kaps-Akad. Forhandlingar, Stockholm, 1865, p. 80, pi. 10, fig. 9 ; Annulata Poly chajta, p. 1-1, 1867. ? Eujiolynoe occidentalis Mcintosh, Annals and Magazine Nat. Hist., IV, vol. xiii, p. 264, pi. 9, figs. 8-13, 1874. This large species is easily distinguished, even when destitute of 38 Smith and Harger — St. George's JBanks Dredging s. its scales, by the short, stout, strongly curved setae of the upper ra- mus, and much longer, slender, fascicled settle of the lower ramus, among which the upper ones have a strongly spinulose, slender, acu- minate, terminal portion, with a nearly straight, split, or forceps-like, slender tip, while the middle and lower ones have a short, cuspidate terminal portion, with few large spinules, and naked acute tips. I am unable to find anything in the figures and description of the species recently described by Mcintosh to indicate that it is distinct from the present species, with which, however, he has not compared it. Near Saint George's Bank, 150 fathoms, mud (locality s). Gulf of Saint Lawrence, 110 fathoms (Whiteaves, t. Mcintosh). — A. E. V. Nephthys circinata Verriii, sp. nov. Body slender, elongated, ratlier depressed, tapering gradually pos- teriorly. Head sub-pentagonal, rather broader than long ; a pair of short, tapering antennae at the anterior angles, about one-fourth as long as the width of the anterior border of the head ; another pair of longer, slender, tapering antenna; at the lateral angles ; tentacular cirri long and tapering. Proboscis smooth towai'd the base ; its dis- tal portion with rows of slender acute papillae, which increase rapidly in length toward the end, where they become very prominent. The lateral appendages, including the setae, are as long as the breadth of the body ; the setae are very numerous, long and slender. The caudal cirrus is long and slender, tapering to a slender tip. Length of body, 50'"'"; diameter, 2-5'"'" ; diameter, including append- ages, 5'"'". The lateral appendages of the middle region are moderately long, the rami separated by a space scarcely equal to half their height. Superior ramus, with a short, broad ovate, obtuse, or slightly acumi- nate upper lamella, directed outward, and considerably exceeding the setigerous lobe, and a much smaller ovate median lamella ; branchial cirrus long, rather slender, tapered, curved downward and inward (cir- cinate), forming rather more than a complete whorl ; the appendage at its base, on the anterior segments, is short and broad, subtruncate dis- tally, and with a small papilliform process projecting downward from its lower angle, nearly in contact with the branchial cirrus ; on the median segments it is broad and long-ovate, unequally acuminate, leaf-like. The lower ramus has a very long and wide ligulate lamella, directed obliquely upward and outward, usually more than twice as long as the setigerous lobe, and about equal to it in width ; its lower edo-e at about the middle is sometimes incurved, and its tip is acumi- nate and blunt-pointed ; the ventral cirrus is slender and tapered. Smith and Harger — St. George's Banks Dredging s. 39 The capillary seta? form large fascicles and are very long and slender, nearly smooth, and with very attenuated tips ; their length is about three times that of the appendages themselves; the transversely marked seta? are scarcely one-fourth as long, and about the same in diameter, with very slender tips. — A. E. V. East of Saint George's Bank, 430 fathoms (locality </) ; north of Saint George's Bank, 85 fathoms, mud (locality jk»). Nephthys ingens Stimpson. SjTiopsis of the Marine Invertebrata of Grand Manan, p. 33, 1853; Verrill, Report on the Invertebrate Animals of Vineyard Sound and Adjacent Waters, in Report of U. S. Commissioner of Fish and Fisheries, part I, IS'ZS, p. 583 (separate copies, p. 289), plate xii, figs. 59, 60, 1874. ? Neplithys incisa Malmgren, Q5fversigt af Kongl. Yet.-Akad. Forhandlingar, 1865, p. 105, plate xii, fig. 21. This is the most common and abundant species on muddy bottoms in the deep water along the whole New" England coast. It occurs at all depths from 2 to 430 fathoms. It is easily distinguished by the stout quadrangular body, deeply incised posteriorly; by the blackish setae, and by the remarkably elongated and widely separated rami of the posterior appendages. There is a long, odd, median papilla on the dorsal side of the proboscis, and a smaller one beneath ; the papillffi in the longitudinal rows are rather small. — A. E. V. Phyllodoce catenula Verrill. Report on the Invertebrate Animals of Vineyard Sound, in Report of U. S. Commis- sioner of Fish and Fisheries, part I, 1873, p. 587, 1874 ; Exploration of Casco Bay by the U. S. Fish Commission, Proceedings American Association for the Ad- vancement of Science, 1873, p. 380, pi. 3, fig. 1, 1874. Plate IV, figure 3. George's Bank, 50 fathoms (locality d). It also occurs at Watch Hill, llhode Island, in 4 to 6 fathoms, among rocks and alga?, and in tide-pools ; at Wood's Hole, at surface, evening, July 3 ; in Cuasco Bay, 8 to 30 fathoms ; and is very common in the Bay of Fundy, from low-water to 50 fathoms. This species is closely allied to P. ptdchella Malmgren, from north- ern Europe, but differs somewhat in the form of the head, which is shorter and rounder in the latter; the branchia? also differ in form. Eusyllis phosphorea Verrill, sp. nov. Plate VII, figure 3. Body slender, elongated, tapering gradually posteriorly. Head, in alcoholic specimens, broader than long, well-rounded in front, the posterior margin incurved ; but in living specimens the head is longer 40 Smith and Hargei — St. George's Banks Dredgings. than broad and slightly narrowed posteriorly. Eyes small, but con- spicuous, wide apart, the anterior considerably farther apart than the posterior ones. Palpi large in presex'ved specimens, broad ovate, and well rounded anteriorly, in contact at their bases; but in living speci- mens more elongated and oblong, exceeding the length of the head. Antennffi (or tentacles) long and slender, distinctly and rather regu- larly annulated, but not moniliform. Tentacular cirri, in preserved specimens, similar to the antennae ; the upper ones are of about the same length, but the lower are little more than half as long. Dorsal cirrus of the second segment is as long as, or even longer than, the antennoe. The dorsal cirri on the 3d, 4th, and 5th segments are shorter, about equal, longer than the lower tentacular cirrus, and about half as long as the dorsal cirri of the succeeding segments, which are alternately longer and shorter, the longer ones about half as long as the breadth of the body. While living, the alternate dorsal cirri are usually held extended and curled up over the back. The two anal cirri are long and slender ; in one preserved specimen they ai-e more than twice the breadth of the body, while in the same specimen the dorsal cirri on the second and third segments preceding the anal one are considerably longer than those on the segments farther for- ward. The setae are all compound, rather long, mostly considerably bent, with a short, acute -triangular terminal piece, which is very distinctly bidentate at the tip. Color of body, when living, deep salmon, or light yellowish orange, with dark brown intestinal line, darker posteriorly ; eyes dark brown. Length, when living, about 25'""'; breadth, rS""". Saint George's Bank, 45 fathoms, among hydroids; Bay of Fundy, off Grand Menan, 52 fathoms, among hydroids. This species, when living, was most brilliantly phosphorescent, with a bright green light, so intense- as to be distinctly visible in daylight, or close to a large kerosene lamp. — A. E. V. Ninoe nigripes Verriii. Report on the Invertebrate Animals of Vineyard Sound, in Report of U. S. Commis- sioner of Fish and Fisheries, part I, 1873, p. 595, 1874; Proceedings American Association for Advancement of Science, 1873, p. 382, pi. 3, fig. 5, 1874. Plate V, figure 3. Locality o, 110 fathoms. Also Fisher's Island Sound, Vineyard Sound, and Buzzard's Bay, and waters outside, in 8 to 29 fathoms, mud; Casco Bay, 10 to 68 fathoms; off the coast of Maine, at various depths to 107 fathoms. Smith and Harger — St. George's Banks Br edgings. 41 Leodice vivida Verriii. Eunice vivida Stimpson, Marine Invertebrata of Grand Manan, p. 35, 1853. Leodice vivida Verrill. American Journal of Science, III, vol. v, p. 9, January, 1873. Plate V, figure 5. Nothria conchylega Maimgren. OnupMs conchylega Sars, Beskrivelsir og lagttagelser, p. 61, pi. 10, fig. 28 {teste Maimgren), 1835. Onuphis Eschrichti (Ersted, Gronlands Annulata Dorsibranchiata, p. 20, pi. 3, figs. 33-41, 45, 1843. Northia conchylega Johnston, Catalogue of British "Worms, p. 138, 1865. Nothria conchylega Maimgren, Annulata Polychseta, p. 66, 1867. Plate VII, figure 3. This species is abundant in the deeper waters, especially upon hard bottoms, on the whole northern coast of Xew England, and in the Gulf of St. Lawrence. Maimgren records it, in 30 to 250 fathoms, fnmi Greenland, Spitzbergen, Finmark, and the coast of Norway. The name " Nothi'ia " was substituted for Northia (Johnston) by Maimgren for reasons that are scarcely sufiicient. The latter name was, however, previously in use for a genus of shells (Gi'ay, 1847), and must be rejected on that account. Nothria opalina Ven-iii. American Journal of Science, III, vol. v, p. 102, 1873. Plate VII, figure 4. Body long and slender, narrowed anteriorly, much depressed and of nearly uniform width throughout most of its length ; the five ante- rior segments much longer than the others. Palpi inferior, rather larcre, hemispherical ; antennne small, ovate, close together, on the front of head. Three central tentacles very long and slender, taper- ing, acute, the basal portion regtdarly annulated and thickened for a considerable distance, beyond which the surface is smooth, with an occasional distant annulation ; the central odd one is somewhat shorter and more slender than the two adjacent ones, which reach to or beyond the 10th segment ; outer pair much shorter, being less than half the length of the central ones. Tentacular cirri small and very slender. Lateral appendages or "feet" of the first six se tigerous segments similar in structure but more prominent than the following ones, from which they also difier in having the ventral cirrus well developed, long and tapering, but shorter and thicker on the first segment than on the five following. Those of the first pair have a stout stalk, which terminates in a small, bhuitly rounded se- tigerous lobe, with a long, slender, subterminal cirrus-like lobe above, Trans. Conn. Acad., Vol. III. 6 August, 1874. 42 Smith and Harger — St. George's Banks Bredgings. longer than the stalk ; dorsal cirrus arising from near the base, longer and more slender than the terminal cirrus ; branchial filament simple, long and very slender, about equalling the dorsal cirrus and united to it above its base; ventral cirrus ovate, tapering, blunt, arising from near the base. The second pair of feet are similar to those of the first, except that in the largest specimens there are two branchial filaments, and the ventral cirrus is longer and more slender. The 3d, 4th, 5th, and 6th pairs have essentially the same structure, but the ventral cirrus becomes gradually longer to the 6th, where it is longer than the stalk and nearly equal to the terminal cirrus. The succeed- ing feet are much shorter ; the ventral cirrus is a mere conical papilla, which soon disappears ; the terminal cirriform lobe becomes smaller and disappears after the 10th pair; the branchial filament becomes larger and longer to the middle region, where it exceeds in length half the diameter of the body, while the dorsal cirrus at the same time becomes smaller and shorter, until it is less than one-fourth the length of the branchia. The setae of the anterior feet consist of slender, acutely pointed, curved ones, mixed with much stouter, blunt pointed compound ones ; farther back there are two fascicles of more slender acute setae, and in the lower bundles a few long, stout, bidentate hooks, with a thin, rounded, terminal expansion. Color, in alcohol, pale yellowish white, but everywhere very bril- liantly iridescent, with opaline lustre and colors. Length, Y5 to 125""" ; diameter, 2*5 to 4""". Common in 110 and 150 fathoms, haiils s and o. It was also dredged in 1873, off Casco Bay, in 30 to 94 fathoms, and on Jeffrey's Bank, in 79 to 105 fathoms. It was also abundant, on muddy bot- toms in deep water, at all the localities in the Gulf of Maine examined by Dr. Packard and Mr. Cooke hi 1873. G-oniada maculata CErsted. Ann. Dan. consp., p. 33, figs. 16, 23, 91, 95, 97, 98 (t. Malmgren). Glycera viri- descens Stimpson, Marine Invertebrata of Grand Manan, p. 53, 1853. North of Saint George's Bank, 110 and 150 fathoms, mud (local- ities o and s) ; Saint George's Bank, 20 fathoms (locality j). Off Casco Bay, 30 to 90 fathoms, mud. Bay of Fundy, 20 to 70 fathoms. Common in the Gulf of Maine, 60 to 100 fathoms. Northern coasts of Europe, from Finmark to Scotland, 10 to 130 fathoms (Malmgren). —A. E. V. Smith and Harger — St. George's Hanks Dredgings. 43 Rhynchobolus capitatus Verriii. Glycera capitata (Ersted, Gronl. Ann. Dorsibranchiata, p. 44, plate VII. figs. 87, 88, 90-94, 96, 99 ; Malmgren, Annulata Polychseta, p. 70, 1867 {non Claparede). This species is furuisliecl with four well-developed jaws, and there- fore belongs to the genus Rynchoholus, as constituted by Claparede. The species without jaws, which he refers to Glycera., must be distinct. Saint George's Bank, 60 fathoms (locality e) ; 20 fathoms (locality j)\ 110 fathoms (locality o) ; east of Saint George's, 430 fathoms (locality g). Greenland, Iceland, Spitzbergen, and northern coasts of Europe to Great Britain. — A. E, V. Samythella VerriU. Body elongated, composed of about 50 segments, 15 of which bear fascicles of setie; and posteriorly about 35 bear uncini only, but have a small conical papilla above the uncigerous lobe, as in Melinna • the uncini commence on the 4th setigerous ring. Branchiae six, placed side by side in a continuous transverse row. Cephalic lobe oblique, somewhat shield-shape, with a narrowed prominent front. Buccal lobe shorter. Tentacles numerous, smooth and slender. This genus is closely allied to Sumytha of Malmgren, in the struc- ture of the head and number of branchiae, but diifers in having a much larger number of segments (in this respect approaching Melinyia)^ and in having only 15 setigerous segments, instead of 17. Samythella elongata Verrili. American Journal of Science, III, vol. v, p. 99, 1873. Body slender, composed of 54 segments in the specimens examined, tapering regularly to the posterior end. Cephalic lobe about as broad as long, broadly rounded posteriorly, with the postero-lateral corners prominent and well rounded, the sides slightly incurved and rapidly narrowing to the front, which is about half the width of the back, and subtruncate, projecting forward ; the middle region is a raised and convex oblong area as wide as the front edge, into which it runs. Buccal lobe a little shorter. Tentacles numerous, slender, tapering. Branchia? subequal, slender, tapering, about twice the length of the cephalic lobe. Setse numerous and long in all the fas- cicles except the first three, the longest nearly one-third the diameter of the body. The posterior end of the body is surrounded by about eight small papilloe, of which the two ujjper ones are largest. Length of largest specimen, in alcohol, 40'"'"; diameter, 2-5 to 3'"'". The tubes consist of a thin and tough lining, to which a close layer of sand, in grains of moderate and nearly uniform size, is firmly cemented. 44 Smith and Marger — ;S'^. George^ Banks Dredgings. G-rymaea spiralis Verriii. Am. Journal of Science, III, vol. vii, p. 407, fig. 2, and plate V, fig. 4, April, 18T4. Plate IV, figure 1. Body long and slender, spirally coiled, composed Fig. i.* of over 150 segments, of which about 120 bear fas- cicles of slender set:*. Branchiae long filiform, two or three times the diameter of body, arising in three clusters on each side, easily detached and often par- tially absent. Setae on the first six or seven seg- ments a little longer than the following ones. Gen- eral color dark red. Tube composed of firmly cemented mud and sand, coiled in a double spiral, the two halves revolving in opposite directions. Also dredged, in 1872, ofi' Grand Menan Island, Bay of Fundy, in 60 fathoms; and in 187;^, off Casco Bay, in 90 fathoms, mud ; and in 80 fathoms on Jeffrey's Bank. ? Potamilla neglecta Maimgren. (Efversigt af Kongl. Vet.-Akad. Forliandlingar, 1865, p. 401, plate 27, fig 84. Sabella neglecta Sars, Reise i Lofot. og Finm., p. 83 (t. Maimgren). This species was very abundant at localities d, h, ^, q, and also occurred in 110 fathoms (locality o). The tubes are long and tough, covered externally with sand. One specimen from Le Have Bank, 45 fathoms (locality A), had a large number of young ones within the tube, adhering to its inner surface. —A. E. V. SpirorblS valida Verrill, sp. nov. Tubes much larger than usual in the genus, round, strong, thick, opaque, white, transversely wrinkled, rather rapidly enlarging, sinis- tral, or coiling in the same direction with the hands of a watch ; in some specimens, found attached to flat shells, the tubes form low, rapidly enlarging spirals of several turns, the last whorl enveloping and concealing the others externally, except near its termination, where it rises obliquely upon the preceding one, but leaving a broad, shallow umbilicus in which the previous whorls are visible ; in other specimens, attached to convex univalve shells {Turritella erosa, etc.), the whorls rest upon the upper side of each of the preceding ones, forming an elevated and often somewhat irregular spiral, increasing in size upward, with a small umbilicus, and usually with the last part of the upper whorl slightly free from the preceding one and ascending * Tube of Grymoea spiralis, natural size. Smith and Harger — St. George's Banks Bredglngs. 45 obliquely. Diameter of the larger tubes, at end, 1-75""" to 2"""; height of the more elevated spirals, 3'""' to 5""". There are 15 large branchiae in the adult specimens : 8 on the left side, 7 on the right, with the operculum ; the pinnae are long, slender, extending to near the ends of the branchiae, which have slender and short, naked tips. Operculum large, Avhite, calcareous, irregularly obconic, obliquely truncated, with the outer surface concave, the dorsal side gibbous, the margin slightly sinuous but entire, except for a small notch, or emargination, in the dorsal edge ; the dorsal por- tion is translucent, while the ventral portion is opaque and contains small, round, ova-like bodies ; the pednncle is rather sliort and stout gradually expanding into the base of the operculum, but swollen in the middle, on the dorsal side. Collar, in the specimens examined, considerably mutilated, apparently with a sinuous but not revolute anterior margin, and with a long posterior dorsal point. The region covered by the collar bears, at least on the left side, three large fjxs- cicles of slender, acute, yellowish seta^, both above and below ; the anterior fascicles are directed forward, and the upper anterior one is larger than the other fascicles. Le Have Bank, 45 and GO fathoms (localities h and ^). The size of this species is exceptionally large, and the branchiae are unusually numerous for the genus Spirorbis, to which I refer it with some hesitation. When living specimens can be studied it may prove to be a new genus. It has, like Vermilia, a calcareous operculum, but in form and structure this organ resembles that of some species of Spiro7'bis. — A. E. V. ? Spirorbis nautiloides Lamarck. Anim. sans Vert., ed. I, vol. v, p. 359, 1818. ? Spirorbis communis Quatrefages, Histoire naturelle des Anneles, vol. ii, p. 489. Plate IV, figure 4. The species figured agrees pretty well with that described by Quatrefages, but may not be the same as that of Lamai-ck, which is regarded by several writers as synonymous with it, and by others with S. horealis, the species so abundant on Facus at low-water mark, on our shores. The present species is seldom, if ever, found at low-water mark, and occurs chiefly on stones and shells in deep water. The tubes are opaque, white, cylindrical, rather closely coiled, the terminal portion not erect, and the surface is more or less conspicuously marked with lines of growth. Abundant on the hard bottoms at Saint George's Bank; Casco 46 Smith and Harger — St. George's Banks Dredging s. Bay; Cashe's Ledge; and in the Bay of Fimdy, 10 to 106 fathoms. — A. E. y. Protula media stimpson. Marine Invertebrata of Grand Manan, p. 30, 1853. Plate VI. This species usually forms much contorted and irregularly bent tubes, which are cylindrical and nearly smooth, but with irregular lines of growth. North of Saint George's Bank, 110 fathoms (locality o). Often brought up by fishermen on Saint George's Bank, attached to shells and stones. Abundant on Cashe's Ledge, 50 to 70 fathoms ; ofi" Grand Menan, 30 to 50 fathoms ; oif Casco Bay. — A. E. V. ? Protula borealis Sars. Vidensk. Selsk. Forhandlinger, 1871, p. 417 (separate copies, p. 14). Numerous empty tubes from the muddy bottoms in 110 and 150 fathoms (localities o and s) diifer considerably in form from those of the P. media., ordinarily met with, and may be this species, if distinct. But they may, very likely, prove to be only a variation of the former, due to the muddy character of the bottom. The tubes are much less bent and contorted, often but slightly curved, or nearly straight, nearly smooth, but with occasional ridges or folds, indicating periods of growth. — A. E. V. Gephteea. Phascolosoma CSementarium Verrill (Quatrefages sp.). American Journal of Science, III, vol. v, p. 99, 1873 ; and Report upon the Inverte- brate Animals of Vineyard Sound, in Report of U. S. Commissioner of Fish and Fisheries, part I, 1873, p. 627, pi. xviii, fig. 92, 1874. Very common on the coast of New England, from Long Island Sound northward, in 5 to 430 fathoms, in dead univalve shells. Phascolosoma tubicola Verriii. American Journal of Science, III, vol. v, p. 99, 1873; Proceedings American Asso- ciation for Advancement of Science, 1873, p. 388, 1874. Body versatile in form ; in contraction short, cylindrical, oval or fusiform, 12 to 25'""' long, 2-5 to 4""" in diameter; in full extension the body is moi-e or less fusiform, gradually tapering anteriorly into the long, slender, nearly cylindrical retractile portion, which is longer than the rest of the body, and bears, near the end, a circle of about ten to sixteen simple, slender tentacles, beyond which the terminal Smith and Hurler — St. George's Banks Dredgings. 47 ])ortion is often extended into a short proboscis, witli the montli at the end ; below the tentacles there is sometimes a dilation, but this is without special spines or granules, and like the rest of the retractile portion in texture. The posterior end of the body is bluntly rounded, and the skin is transversely wrinkled and rough, and covered with small, round, somewhat raised verrucas or suckers, to which dirt ad- heres, and at the end nearly always bears from 3 to 8 small, but prominent, peculiar bodies, having a slender pedicle and a clavate or globular head ; their nature is doubtful. (They may be sense-organs, but should be examined on living s])ecimens.) At about the poste- rior third of the proper body is an irregular zone of numerous, dark brown, hard chitiuous hooks, arranged in several rows, broad triangu- lar in form, with acute points directed forward ; among the hooks are also a few suckers ; the middle region is covered with small, round, slightly raised suckers, which become much more prominent and crowded at the anterior end toward the base of the retractile portion, and have here the form of small, subconical, elevated warts, to which dii't usually adheres firmly ; the retractile portion is covered through- out with minute conical verruca? or paj^illte, most prominent toward the base. In many respects P. cmmentarium agrees very closely with this, but it has the posterior end much smoother, and with less conspicu- ous suckers ; the hooks are not so numerous, less acute, and lighter colored; the anterior part of the body has smaller and less j^rominent suckers or A'erructe ; the skin is lighter colored, thinner, and more translucei^t, and there is a zone bearing several rows of minute, slen- der, acute, chitinous spinules, a little l>elow the tentacles. Haiils^^ o, and s, 85 to 110 fathoms. It has also been dredged, in 60 to 94 fathoms, off Casco Bay. ? Phascolosoma boreale Keferstein. Beitrage zur Anat. und syst. Kentniss der Sipimculiden, p. 206. This species is rather short and thick, obtuse posteriorly, nearly smooth to the naked eye, and destitute of both hooks and distinct suckers, but the skin is minutely wrinkled transversely, and covered with almost microscopic slender papillte, and is minutely specked with dirty yellowish brown ; the retractile i)ortion is more distinctly granulated anteriorly. The tentacles are rather numerous, small, and simple. Dredged also off Casco Bay, 64 fathoms; Cashe's Ledge, 50 to 72 fathoms ; and iii the Gulf of St. Lawrence (Whiteaves). 48 Smith and Harger — St. George's J^anks Dredgings. MOLLUSCA. Pleurotomella Packardii VerriU. American Journal of Science, III, vol. v, 1873, p. 15 (December, 1872). Shell thin, fragile, translucent, pale flesh-colored, moderately stout, with an acute, somewhat turreted spire. Whorls nine ; the apical whorls, for about two and one-half turns, are nearly smooth, regular, convex, chestnut-colored ; below this the whorls are shouldered, strong- ly convex in the middle, but with a smooth concave band below the suture, corresponding to the posterior notch in the outer lip; the whorls are crossed below the sub-sutural band by about 16 strong, prominent, rounded, somewhat oblique ribs, most prominent on the middle of the whorl, but not angulated ; on the last whorl these ribs become very oblique below the middle, and follow the curve of the edge of the lip, nearly fading out anteriorly ; the surface between the ribs is marked by faint lines of growth and by fine, unequal, slightly raised revolving lines, which pass over the ribs without intei-ruption. They become more evident on the lower part of the last whorl, and are very faint on the sub-sutural band, which is more decidedly marked by receding, strongly curved lines of growth. The aperture is rather broad above, elongated below, sub-oval, outer lip very thin, sharp, prominent above, separated from the preceding whorl by a wide and very deep sinus, extending back for about one-fifth of the circumfer- ence of the whorl ; the anterior border of the lip is incurved near the end, and obliquely truncate, forming a short, straight canal. Colu- mella simple, nearly straight, its inner edge toward the end sharp, and obliquely excurved. No operculum. Length, 21 •2""" ; breadth, 11-2'""'; length of aperture, 120™'"; breadth of same, 5-0""". The absence of eyes and operculum, great size of the posterior sinus, and character of the apex, indicate that this shell represents a new genus. One living specimen from (o) 110 fathoms. Ringicula nitida Verriii. American Journal of Science, III, vol. v, 1S73, p. 16 (December, 1872). Plate I, figure 2, Shell small, white, smooth, broad oval, with five Avhorls, spire rap- idly and regularly tapered, sub-acute, shorter than the aperture. Whorls very convex, regularly rounded, the sutures well impressed ; a well marked, impressed, revolving line just below the suture ; the surface otherwise nearly smooth, but with more or less distinct, distant, microscopic revolving lines, most distinct anteriorly. Aj^er ture somewhat crescent-sha})ed. Outer lip evenly rounded, forming Smith and Harger — St. George's Banks Dredging s. 49 the segment of a circle, the border regularly thickened, receding a little posteriorly, near the suture. Callus on the body whorl narrow, nearly even, but a little swollen in the middle and slightly raised. Columella stout, recurved at the end, with two strong, very promi- nent, equal, spiral folds — tlie anterior one projecting beyond the canal, with the end rounded. Length, 4-2""" ; breadth, S'l""" ; length of aperture, 2-5 """ ; breadth of aperture, -11""", From 110 and 150 fatlioras (localities s and o). Torellia vestita Jeffreys. This shell in form and size somewhat resembles large specimens of Margarita helicina, but it has a ciliated epidermis resembling that of Veliitina laevigata. The spire is small and low; whorls four, the last large, well rounded, forming the bulk of the shell. Suture deep. Umlnlicus small and deep, somewhat concealed by the reflected outer edge of the columella, which recedes in front and joins the outer lip at an obtuse angle, forming a broad, shallow, anterior emargination ; inner border of the columella a little excavated near the body whorl, slightly swollen in the middle. Outer lip sharp, regularly rounded. Epidermis thick, greenish, with conspicuous lines of growth, finely reticulated by raised revolving lines, along which arise numerous slender, but short, hair-like processes. Shell beneath the epidermis white, nearly smooth. Length, 7*5™"^; breadth, 10™™; length of aperture, 6'"™ ; breadth, 4-5""". Only one specimen, dead and inhabited by a Phascolosoma, was found in 1872. Since this, however, during the explorations of 1873, it was dredged by Dr. Packard and Mr. Cooke, in 52 to 90 fathoms, on Cashe's Ledge, ofl:" the coast of Maine. Stylifer Stimpsonii Verriii. American Journal of Science, III, vol. iii, p. 283, 1872. Plate I, figure 1. Shell white, short, swollen, broad oval; spire short, rapidly enlarg- ing. Whorls four or five, the last one forming a large part of the shell ; convex, rounded, with the suture impressed ; surface smooth, or with faint striae of growth. Color, when living, pale orange yel- low. Length, about 4™'" ; breadth, 3"'". Parasitic on Strongylocentrotus Drobachiensis. In 32 fathoms oft' the coast of New Jersey (Capt. Gedney) ; 60 and 65 fathoms (e and /), George's Banks ; 8 fathoms off" Fisher's Island, mouth of Long Island Sound. Trans. Conn. Acad., Yol. III. 7 August, 1874. 50 Smith and Harger — St. George's Banks Dr edgings. Astarte undata Gould. VerriD, American Journal of Science, III, vol. iii, pp. 213, 287, 1872; and Report on the Invertebrate Animals of Vineyard Sound, in Report of U. S. Commissioner of Fish and Fisheries, part I, 1873, p. 384, pi. 29, fig. 203, 1874. Plate I, figures 6 to 9. The figures given in Gould's works are scarcely characteristic of this, the most abundant species of the northern coast of New England, and we here publish several figures, prepared by Professor Verrill, which more fully illustrate the different forms of the species. The name undata was proposed by Gould for a form of his Astarte sid- cata. Astarte lens Stimpson. Astarte crebricostata G-ould, Invertebrata of Massachusetts, 2d edition, edited by Binney, p. 126, fig. 440, 1870 (not of Forbes, teste Verrill). Astarte lens Stimpson, MS., Gould, op. cit., p. 127 ; Verrill, American Journal of Science, III, vol. iii, pp. 213, 287, 1872. Plate I, figures 4 and 5, This species seems to be more exclusively a deep-water form than the last, although the specimens dredged by us at the localities {g, o, and s) mentioned are all much smaller than the common form of the species in the Bay of Fuiuly, and may well be regarded as a dwarf variety. Pecten pustuloSUS VerriU. American Journal of Science, III, vol. v, 1873, p. 14 (December, 1872). Upper valve more convex than the lower, a little swollen toward the umbo ; length and breadth nearly equal, the margin diverging nearly at right angles from the beak to the middle of the anterior and posterior borders, on each of which tliere is an obtuse angle, from which the outline of the ventral margin forms a regular curve, nearly semicircular, but a little produced ventrally ; the surface with about 14 radiating rows of relatively large, prominent, round, hollow vesi- cles, those in the middle rows nearly hemispherical, while part of those of the lateral ones ai"e subconical and smaller ; seven or eight of the rows are first developed, at a short distance from the apex of the shell, the other ones afterward coming in between the primary ones ; the rows are distant in the middle and more crowded together toward the borders; between the rows of vesicles the surface is marked by distant, fine, impressed grooves, which pass between and separate the vesicles ; on the umbos, above the origin of the vesicles, the border of the groove rises into a thin, slightly elevated lamella. Lower valve with fine, close, slightly raised, concentric lamellae, be- Smith and Harger — St. George's Banks Dredgings. 5 1 coming faint toAvard the beak. Auricles unoqiuil,,tliat of the upper valve small, and a little projecting posteriorly, much longer and more prominent, with a deep, curved emargination anteriorly, its surface with concentric lamellae and radiating rows of small, conical vesicles ; that of the lower valve with a deep, angular byssal notch anteriorly, its surface with concentric lamelhi3 and faint radiating ridges. Color yellowish white. Length, 7-5""" ; height, 8-0""" ; thickness, 2-5""", East of St. George's Banks {g), in 430 fathoms, dead but fresh valves; and north of the Banks, locality {s), 150 fathoms, living. Pera crystallina VerriU. Clavelina crystallina Moller, Naturliistorisk Tidsskrift, vol. iv, p. 95, 1842. Pera imllucida Stimpson, Proceedings Boston Soc. Nat. Hist., vol. iv, p. 232, 1852. Pera crystallina Verrill, American Journal of Science, III, vol. iii, p. 213, pi. 8, fig. 9, 1872. Plate VIII, figure 1. This species was described by Stimpson from specimens, adhering to stems Sertidarelkt polyzonias, variety gigantea, taken in 30 fathoms on St. George's Banks. Professor Verrill records it from Murray Bay, Gulf of St. Lawrence. Glandula arenicola Verrill. American Journal of Science, III, vol. iii, pp. 211, 288, 1872 ; Report on the Inver- tebrate Animals of Vineyard Sound, in Report of U. S. Commissioner of Fish and Fisheries, 1873, p. 701, 1874. This species, which was dredged by us in immense numbers in 28 fathoms (haul c), has also been dredged, by Dr. Dawson, at Murray Bay, Gulf of St. Lawrence, by Mr. T. M. Prudden, in Buzzard's Bay, and off New London, Conn., by A. E, Verrill. Thyone scabra Verriii. American Journal of Science, 111, vol. v, p. 100, 1873. Thyone fusus ? Verrill, American Journal of Science, III, vol. v, p. 14, 1873 (mow Koren). Body fusiform, gi-adually tapered behind, with a long, slender, pos- terior portion, covered throughout with very numerous, rather rigid, slender, scabrous papillae ; skin rather rigid, scabrous with small, rough points, which project from the plates. Tentacles ten ; eight large ones much elongated and arborescently divided from near the base ; the two small ones are very short, nearly sessile, subdivided from the base. The calcareous plates of the skin are very flat, some- what imbricated, irregularly oval, triangular, or subpolygonal, with an undulated or crenulated margin, pierced by about 20 to 24 unequal round openings, tAvo or three central ones larger than the rest, the 52 Smith and Harger — St. George's BanJcs Dredgings. interspaces mostly.as wide as the pores ; from the center of the upper side arises an open, slender, flat, acute spinous process, composed of two anastomosing pieces. The plates of the papillae or suckers are narrow, elongated, bent into a bow-shape, the middle expanded and usually pierced by about four pores, two of which are larger ; the ends are also usually dilated and pierced with small pores ; from the middle arises a flat, spinous process, similar to that of the skin-plates, but smaller. Length, in alcohol, about 50""" ; greatest diameter, 6 to 9™™ ; length of longest tentacles, 7"5™™. Color of pi-eserved specimens, yellowish brown. Localities o and s, 110 and 150 fathoms. Also dredged, in 1873, oflT Casco Bay. This species resembles T. raphanus Duben and Koren (Troschel sp.) in form, but the latter has long-stalked tentacles, branching only near the ends, and the plates of the skin are different in form, and in the perforations, and lack the spinous processes which give the species its rough, scabrous surface, ? Charybdea periphylla Peron and Lesueur. Verrill, Report upon the Invertebrate Animals of Vineyard Sound, p. 724, 1874. This species, originally described and figured by Peron and Lesueur from mutilated specimens taken under the equator in the Atlantic Ocean, is doubtfully identified b)' Professor Verrill with a specimen obtained by us east of George's Banks. The body in the alcoholic specimen is elevated, bell-shaped, rounded above, with a marked constriction toward the border ; transparent, the inner cavity showing through as a large, conical, dark reddish brown spot, with the apex slightly truncated. Border dcejily divided into sixteen long, flat lobes, which are of nearly uniform breadth throughout, and slightly rounded, or sub-truncate, at the end ; the edges and end thin and more or less frilled ; the inner side with two sub-marginal carinae. Eyes inconspicuous, but small bright red specks are scattered over the marginal lobes. The intervals between the lobes are narrow and generally smoothly rounded, without dis- tinct evidence of the existence of tentacles, except that, in one of these intervals, there is a small and short papilliform process, with brown pigment at the base. The ovaries are mostly wanting, but portions are to be seen as slightly convoluted organs in the mar- ginal region, opposite the intervals between the lobes. Smith and Harger — St. George's Banks Dredging s. 53 Lafoea gracillima G- 0. Sars (Alder sp.). Lafoea fruticosa Hincks, History of British Hydroid Zoophytes, p. 202, pi. 41, fig. 2, 1868; and Annals and Magazine Nat. Hist, IV, vol. xiii, pp. 132, 148, pi. G, figs. 6-10, pi. 7, fig. 16. 1874. Lafoea gracillima G. 0. Sars (Alder sp.), Bidrag til Kundskaben om Norges Hydroi- der, in Vidensk.-Selskabs Forhandlinger, Christiania, for 1873, p. 115 (27), pi. 4, figs. 19-21. Hincks reports this species from 100 fathoms off the coast of Iceland, and G. O. Sars from a depth of 150 fathoms off the Norwegian coast. It has been dredged by Professor Verrill in the Bay of Fundy and in Casco Bay. Halecium robnstum Verrill. American Journal of Science, III, vol. v, 1873, p. 9, December, 1872. Stem stout and coarse, composed of many tubes ; branches stout, tapering, compound except at tips, pinnately or bipinnately branched, the branchlets spreading at an angle of aboiit 45° ; yellowish white and translucent, about '5 of an inch long, divided by simple distant constrictions, the long internodes usually bearing from two to four hydroids. Hydrothecoe alternate, large, deep, somewhat vase-shaped, with an even, slightly evei'ted rim, below which there is a slight con- striction ; the middle region is slightly smaller, gradually narrowed toward the base, with a simple diaphragm near the base within. The hydrothecJB are articulated upon slightly prominent projections from the stem, in an oblique and excentric position, so as to produce a decidedly geniculated appearance. Most of the hydrothecse are sim^jle, but some have one or two slightly pi'ominent secondary rims near the margin. Height about 100'"™. East of St. George's Bank, 430 fathoms (haul g). Sertularella polyzonias Gray, var. gigantea Hincks. Annals and Magazine Nat. Hist, IV, vol. xiii, p. 151, pi. 7, figs. 11, 12, 1874. Diphasia mirabilis Verriii. American Journal of Science, III, vol. v, 1873, p. 9, December, 1872. Stem stout, rather rigid, narrowed at base, pinnately branched, somewhat flexuous between the branches, which are alternate, stout, rigid, straight, constricted at base, spreading at an angle of about 45°. Hydrothecae on the main stem in two rows, nearly opposite ; on the branches mostly in six regular rows, occupying all sides of the branches, those in the adjacent rows alternating. The hydrothecae have large, appressed, somewhat swollen bases, but the upper portion is rapidly narrowed, prominent and curved outward; aperture strongly bilabiate, operculated. Reproductive capsules not observed. Le Have Bank, 60 fathoms (haul e). 64 Smith and Harger — St. George's Banks Dredgings. Pennatula aculeata Danieisen. Pennatula aculeata Danieisen, Forhandlinger i Vedenskabs-Selskabet i Cliristiania, 1858, p. 25 (teste Kolliker) ; VerriU, loc. cit., p. 100, 1873. Pennatula phosphorea, var. aculeata, Kolliker, Anatomisch-systematische Beschrei- bung der Alcyonarien, 1 Abtheilung, 1 Halfte, p. 134, pi. 9, fig. 73, 1870 (from Ahandlungeu d. Senckenberg. Naturf. Gesellschaft. Frankfort, Bd. vii). Pennatula Canadensis Whiteaves, Annals and Magazine of Natural History, IV, vol. X, p. 346, November, 1872. Pennatula, near P. phospliorea VerriU, Am. Journal of Science, III, vol. v, p. 5, 1873. Localities o and s, 110 and 150 fathoms. Also dredged by Mr. Whiteaves in 200 fathoms in the Gulf of St. Lawrence. Virgularia Lyungmanii Kolliker. Op. cit, 2 HaKte, 1 Heft, p. 196, pi. 13, figs. 133, 134, 1871; Verrill, American Journal of Science, III, vol. v, p. 100, 1873 ; "Whiteaves, Report on a Second Deep-sea Dredging Expedition to the Gulf of St. Lawrence, p. 13, 1873. This species was described by Kolliker from specimens obtained in 30 to 80 fathoms, among the Azores, by the Josephine Expedition sent ont by the Swedish government. It was also dredged in 1872, in the Gulf of St. Lawrence, at a depth of 200 fathoms, by Mr. Whiteaves. Urtlcina nodosa Verriii. Actinia nodosa Fabricius, Fauna Groenlandica, p. 350, 1780. Urticina digitata Verrill, Am. Jour, of Science, III, vol. v, p. 5, 1873 (not of MiiUer?). This species has been dredged also in deep water off" Casco Bay (Professor Verrill), and in the Gulf of St. Lawrence (Mr. Whiteaves). Cerianthiis borealis Verrill. American Journal of Science, III, vol. v, 1873, p. 5, December, 1872. Plate II, figure 5. Body much elongated, tapering gradually to the abactinal opening, the surface smooth but more or less sulcated longitudinally. Marginal tentacles very numerous and unequal, the inner ones longest, in the largest specimens 56'"'" long, and 3™"' in diameter at base, gradually tapering, acute ; the outer ones 25'""' and less in length. Oral tentacles numerous, crowded in several rows, in the largest specimens about 25""" long, slender, acute. Color of body olive-brown or dark chestnut-brown, sometimes pale bluish or purplish just below the tentacles ; disk pale yellowish-brown ; space within the oral tentacles, around the mouth, deep brown, witli lighter radiating lines ; oral tentacles pale chestnut-brown ; marginal ones deep salmon or yellowish-brown, the longest usually barred tr^s- versely with six to eight dark reddish-brown spots, each spot partially divided along the median line into two lateral ones ; part of the tenta- cles often have flake white spots on each side, at the base. Smith and Harger — St. Georges Banks Dredglngs. 55 The two largest specimens, dredged in 1872, in 28 fathoms, east of Grand Menan, by Professor Verrill, measured 125""" across the disk and tentacles, but their bodies were mutilated. Entire ones of much smaller size were dredged by Dr. Packard and Mr. Cook in 110 and 150 fathoms, soft muddy bottom, hauls ^s' and o. The largest of these was 200'""' long, and like other species of the genus, iidiabited a thick, tough, felt-like, muddj^ tube. It was also dredged, in 1873, in Casco Bay, from 7 to 94 fathoms. One of these speci- mens, dredged off Seguin Island, in 70 fathoms, was 450'"'" long, 40""" in diameter, and 175'"'" across the tentacles. A small specimen has been dredged in 18 fathoms off Watch Hill, R. I. Epizoanthus Americanus Verriii. Plate VIII, figure 2. This species lives upon stones as well as upon shells inhabited by Eupagiiriis. The specimens from 430 fathoms {g) were on stones, while those from 60 and 65 fathoms {s and /') were on shells. It ranges from off the coast of New Jersey to the Gulf of St. Lawrence. Sponges. Most of the sponges obtained have not yet been sufficiently studied to be reported upon, but the two following species are of special interest. Hyalonema longissimum Sars. G. 0. Sars, on some Remarkal^le Forms of Animal Life from the Great Deptlis off the Norwegian Coast, p. TO, pi. 6, figs. 35-i5, 1872. Only a single and somewhat abnormal specimen of this remarkable species was dredged by us in 430 fathoms, but it has since been dredged in considerable abundance by Professor Verrill, in 95 fathoms, off Casco Bay, and by Dr. Packard and Mr. Cooke on Cashe's Ledge.* Mr. Whiteaves reports it also from deep water in the Gulf of St. Lawrence. TheCOphora ibla WyvlUe Thompson. Depths of the Sea, p. 147, fig. 2-i, 1873; Verrill, American Journal of Science, III, vol. vii, p. 500, pi. 8, fig. 8, 1874. Plate VII, figure 1. This species, first described by Wy ville Thompson, from specimens dredged in 344 fathoms, off the Shetland Islands, l)y the Porcupine expedition, and dredged by us in 50 and 60 fathoms (hauls e and d), has since been dredged by Dr. Packard and Mr. Cook on Cashe's Ledge and Jeffrey's Ledge in the Gulf of Maine. * American .Tournal of Science, III, vol. vi, p. 440, 1873. 56 Smith mid Sarger — St. George's Batiks Dredging s. EXPLANATION OF PLATES. Plate I. Figure 1. — Stylifer Stimpsonii Verrill ; specimen from 60 fathoms, George's Bank (haul i) ; enlarged 10 diameters. Figure 2. — Ringicula nitida Verrill ; specimen from 110 fathoms; enlarged 14 diameters. Figure 3. — Entalis striolata Stimpson ; several views of animal, with the foot in differ- ent states of expansion ; enlarged about 1^ diameters. Figure 4. — Astarte lens Stimpson ; adult ; natural size. Figure 5. — The same ; young specimen ; natural size. Figure 6. — Astarte undaia Gould ; inside of valves, showing the hinge ; natural size. Figure 7. — The same ; young specimen ; natural size. Figure 8. — The same ; adult specimen ; natural size. Figure 9. — Variety of the same ; adult specimen ; natural size. Figure 10. — Astarte elliptica (Brown) ; natural size. Figure 11. — Cryptodon obesus Verrill; inside of valve ; enlarged 3 diameters. Figure 12. — Astarie Baiiksii lie&ch. ; natural size. Figure 1 was drawn from nature by S. I. Smith; 2, 5, 6, 7, 8, 9, 11, by Professor Verrill ; 3, by J. H. Emerton : the rest from Binney's Gould. Plate II. Figure 1. — Sertularia argentea Ellis and Solander ; a branch bearing reproductive capsules (gonothecse) with the soft parts removed ; much enlarged. Figure 2. — Alcyonium carneum Agassiz ; three of the polyps fully expanded ; much enlarged. Figure 3. — Crista eburnea Lamouroux ; a cluster of branches, enlarged. Figure 4. — The same ; a branch bearing ovicells, more highly magnified. Figure 5. — Gerianthus borealis Verrill; entire animal removed from its tube and fully expanded ; about one-third natural size. Figures 1 and 2 were drawn from nature by Professor Verrill ; 3 and 4 by Profes- sor A. Hyatt ; 5 by J. H. Emerton. Plate III. Figure 1. — Xenodea megachir Smith, male ; one of the first pair of legs with its epime- ron, seen from the outside ; enlarged 20 diameters. Figure 2. — The same ; one of the second pair of legs, seen in the same position and enlarged the same amount. Figure 3. — The same ; one of the third pair of legs, with its epimeron and gill, seen from the outside, and showing the glandular organ within ; enlarged 20 diameters ; a, the tip of the dactylus, showing the perforation ; enlarged 100 diameters. Figure 4. — The same ; one of the fifth pair of legs, with its epimeron and gOl, seen from the outside ; enlarged 20 diameters. Figure 5. — Stenothoe peltata Smith, female ; one of the second pair of legs, with its epimeron, seen from the outside ; enlarged 16 diameters. Figure 6. — The same ; one of the fourth pair of legs, with its epimeron, seen from the outside ; enlarged 16 diameters. Figure 7. — The same; one of the first pair of legs, seen from the outside ; enlarged 50 diameters. Figure 8. — The same ; distal portion of the propodus, with the dactylus, of one of the second pair of legs, seen from the outside ; enlarged 125 diameters. Figure 9. — ScaJpellum Stroemi Sars ; side view ; enlarged 5 diameters. All the figures were drawn on wood, from alcoholic specimens, by S. I. Smith. Smith and Harger — St. George's Banks Dredgings. 57 Plate IV. Figure 1. — Gnjmcea spiralis Yerrill; head and anterior part of body ; enlarged. Figure 2. — Pista cristata Afalmgren ; head and anterior part of l)ody ; enhirged. Figure .'i. — Phyllodoce catemda Verrill ; dor.'^al view of anterior part of body and liead, and extended proboscis : enlarged about 4 diameters. Figure 4. — ? SpirorUs nautiloides Lamarck ; entire animal ; much enlarged. All the figures were drawTi from life hj J. H. Emerton. Pl.\te V. Figure I — Nephthys ciliata Rathke ; one of the lateral appendages ; enlarged 10 diameters. Figure 2. — Lumbriconereis fragilis ffirsted ; anterior part of body and head, dorsal view ; enlarged about G diameters. Figure 3. — Ninoe nigripes Verrill ; one of the lateral appendages from the middle })art of the body ; greatly enlarged. Figure 4. — Ammochares assimilis Sars ; entire animal ; enlarged about 4 diameters. Figure 5. — Leodice vivida\QVT\]\] head and anterior part of the body and 12th seg- ment ; dorsal view ; enlarged about 4 diameters. Figure 1 was copied from Ehlers ; all the others were drawn from nature by J. H. Emerton. Plate VI. Protula media Stimpson ; animal removed from the tube ; enlarged 4 diameters. Drawn from life by J. H. Emerton, from a specimen dredged near Grand Menan, Bay of Fundy, by Professor Verrill, in 1872. Plate VII. Figure 1. — Thecophora ibla W. Thompson; specimen from 60 fathoms, Le Have Bank (haul i) ; natural size. Figure 2. — Eusyllis iJhosplwrea Verrill ; anterior and posterior portions of the animal ; dorsal view ; much enlarged. Figure .3. — Nothria conchylega Malmgren ; anterior portion ; enlarged. Figure 4. — Nothria opalina Verrill ; anterior portion ; enlarged. Figure I was drawn from nature by Sherman ; the others from life by J. H. Emerton. Plate VIII. Figure 1. — Pera crystallina Verrill ; enlarged 3 diameters. Figure 2. — Epizoanthus Americanus Verrill ; a single polyp expanded ; enlarged about 6 diameters. Figure 3. — Chcetoderma nitidulum Loven ; entire animal ; enlarged 4 diameters. Figure 4. — The same ; posterior portion with the gills expanded ; enlarged 24 diame- ters. Figure 1 was drawn from nature by Professor Verrill ; the others were drawn from life by J. H. Emerton. ERRATA. Page 1, line 13, for Hagerman, read Hagenman. 13, " 34, " capypilare, read capillare. " 28, " 19, '• Caridon, read Caridion. 35, " 3, " Scapellum, read Scalpellum. " 58, " 14, •' branches, read branchlets. " 60, " 12, ■■' Plate X. read Plate IX. 56 Smith and Sarger — St. George's Banks Dredgings. EXPLANATION OF PLATES. Plate I. Figure 1. — Stylifer Stimpsonii Verrill; specimen from 60 fathoms, George's Bank (haul i) ; enlarged 10 diameters. Figure 2. — Ringicula nitida Verrill ; specimen from 110 fathoms ; enlarged 14 diameters. Figure .3. — Entalis striolata Stimpson ; several views of animal, with the foot in differ- ent states of expansion ; enlarged about 1^ diameters. Figure 4. — Astarte lens Stimpson ; adult ; natural size. Figure 5. — The same ; young specimen ; natural size. Figure 6. — Astarte undata Gould ; inside of valves, showing the hinge ; natural size. Figure 7. — The same ; young specimen ; natural size. Figure 8. — The same ; adult specimen ; natural size. Figure 9. — Variety of the same ; adult specimen ; natural size. Figure 10. — Astarte elliptica (Brown) ; natural size. Figure 11. — Cryptodon ohesus Verrill ; inside of valve ; enlarged .3 diameters. Figure 12. — Astarte Banksii Leach ; natural size. Figure 1 was drawn from nature by S. I. Smith; 2, 5, 6, 7, 8, 9, 11, by Professor Verrill ; 3, by J. H. Emerton ; the rest from Binney's Gould. Plate II. Figure 1. — Sertularia argentea Ellis and Solander ; a branch bearing reproductive capsules (gonothecse) with the soft parts removed ; much enlarged. Figure 2. — Alcyonium carneum Agassiz ; three of the polyps fully expanded ; much enlarged. Figure 3. — Crista eburnea Lamouroux ; a cluster of branches, enlarged. Figure 4. — The same ; a branch bearing ovicells, more highly magnified. Figure 5. — Cerianthus horealis Verrill ; entire animal removed from its tube and fully expanded ; about one-third natural size. Figures 1 and 2 were drawn from nature by Professor Verrill ; 3 and 4 by Profes- sor A. Hyatt ; 5 by J. H. Emerton. Plate III. Figure 1. — Xenoclea megachir Smith, male ; one of the first pair of legs with its epime- ron, seen from the outside ; enlarged 20 diameters. Figure 2. — The same ; one of the second pair of legs, seen in the same position and enlarged the same amount. Figure 3. — The same ; one of the third pair of legs, with its epiraeron and gill, seen from the outside, and showing the glandular organ within ; enlarged 20 diameters ; a, the tip of the dactylus, showing the perforation ; enlarged 100 diameters. Figure 4. — The same ; one of the fifth pair of legs, with its epimeron and gill, seen from the outside ; enlarged 20 diameters. Figure 5. — Stenothoe peltata Smith, female ; one of the second pair of legs, with its epimeron, seen from the outside ; enlarged 16 diameters. Smith and Haryer — St. George's Banks Dredgings. 5V Plate IV. Figure 1. — Gryrmm spiralis Verrill; liead and anterior part of body ; enlarged. Figure 2. — Pista cristata Malmgren ; head and anterior part of body ; enlarged. Figure :5. — Phyllodoce catenula Verrill ; dorsal view of anterior part of body and head, and extended proboscis : enlarged about 4 diameters. Figure i. — ? Spirorbis nautiloides Lamarck ; entire animal ; much enlarged. All the figures were drawn from life by J. H. Emerton. Platb V. Figure 1 — Nephthys ciliata Rathke ; one of the lateral appendages ; enlarged 10 diameters. Figure 2. — Lumbriconereis fragilis (Ersted ; anterior part of body and head, dorsal view ; enlarged about G diameters. Figure 3. — Nino'e nigripes Verrill ; one of the lateral appendages from the middle part of the body ; greatly enlarged. Figure 4. — Ammochares assimilis Sars ; entire animal ; enlarged about 4 diameters. Figure 5. — Leodice vivida Verrill ; head and anterior part of the body and 1 2th seg- ment ; dorsal view ; enlarged about 4 diameters. Figure 1 was copied from Ehlers ; all the others were drawn from nature by J. H. Emerton. Plate VI. Protula media Stimpson ; animal removed from the tube ; enlarged 4 diameters. Drawn from life by J. H. Emerton, from a specimen dredged near Grand Menan, Bay of Fundy, by Professor Verrill, in 1872. Plate VII. Figure 1. — Thecop}io7-a ibla "W. Thompson ; specimen from 60 fathoms, Le Have Bank (haul i) ; natural size. Fjgure 2. — Eusyllis pliosphorea Verrill ; anterior and posterior portions of the animal ; dorsal view ; much enlarged. Figure 3. — Nofhria conchylega Malmgren ; anterior portion ; enlarged. Figure 4. — Nbthria opalina Verrill ; anterior portion ; enlarged. Figure I was drawn from nature by Sherman ; the others from life by J. H. Emerton. Plate VIII. Figure 1. — Pera crystalUna Verrill ; enlarged 3 diameters. Figure 2. — Epizoanthus Americanus Verrill ; a single polyp expanded ; enlarged about 6 diameters. Figure 3. — Chcetoderma nitidulum Loven; entire animal; enlarged 4 diameters. Figure 4. — The same ; posterior portion with the gills expanded ; enlarged 24 diame- ters. Figure 1 was drawn from nature by Professor Verrill ; the others were drawn from life by J. H. Emerton. Errata. 5, line 30, for Vetumnus, read Vertumnus. " 9, last line, for 1873, read 1874. " 11, line 31, for virticillata, read verticillata. Trans. Conn. Acad., Vol. III. 8 July, 1875. II. — Descriptions of New and Rare Species of Hypeoids from THE New England Coast. By S. F. Clark/. A The material for this paper was gathered while at work on the Hydroids hi the Museum of Yale College. This colleetion is mostly from the New England Coast, and is very large and complete. Obelia bictispidata, sp. nov. Plate IX, fig. 1. The stem is erect, slender, straight or nearly so, compound, con- sisting of many united tubes Avhich gradually diminish in number toward the top, varying in color from a light horn, to a light whitish brown, sparingly branched, and with three or four annulations just above the origin of each • branch ; branches short, ascending, slender and irregularly arranged, sometimes one, and often two branches starting from a node ; branches few, very short, slender and ascending. Hydrothecre very deeply cainpainilate, narrow, tapering slightly toward the base, very hyaline, and with eight to ten longitudinal lines extending from the distal extremity nearly to the base ; the rim is armed with very acute teeth, varying in number from sixteen to twenty- two, and arranged in pairs, the spaces in which the longi- tudinal lines terminate being a trifle wider and deeper than the alter- nate spaces ; the pedicels supporting the hydrothecse are long and tapering, consisting of about fifteen annulations. Gonothecfe un- known. Height of largest specimens, about three inches (80"'"^). The specimens from which this species is described were taken in 3-5 fathoms, on the reefs near Thimble Islands, Long Island Sound, Sep- tember 23, 1874. This species is closely allied to 0. Mdentata, but is readily distin- guished from the latter by its entirely diiFerent habit, the narrower and deeper calycles, and by the long tapering pedicels upon which the calycles are supported. Obelia bidentata, sp. nov. Plate IX, fig. 2. Stems clustered, straight or slightly flexuous, compound, composed, at the base, of eight vor ten slender, united tubes, varying in color from a light horn to a dingy wdiite, densely branched, and with three 6\ F. Clark on Neio and Rare Species of Hydro ids. 59 or four auimliitions just al)ove the origin of each branch; mode of branching irregular, two branches often starting from a node, some- times an alternate arrangement of branches on opposite sides of the stem, one branch at each node. A few o\ the lowest branches some- times attain a considerable length and resemble the main stem, the upper ones are short, sparingly branched and with the pinnae diverg- ing at a slight angle ; the branchlets and ends of the branches are simple, slender, translucent, and very graceful. Ilydrothec* very deeply canipanulate, tapering slightly toward the base, and with nine to twelve longitudinal lines extending from the distal extremity nearly to the base; the rim is ornamented with from eighteen to twenty-four very aciite teeth, arranged in pairs, the spaces in which the longitudinal markings terminate being a trifle wider and deeper than the alternate spaces ; the pedicels supporting the hydrothecae are usually short and stout, consisting of three to six strong rings, but some of the hydrothecae near the base of the stem have the pedicel slightly tapering, and composed of from ten to twelve annulations. GonotheciB unknown. The largest specimen has a height of about 6 inches (150'"'"). We have had this species from but one locality, Greenport, Long Island, where it was collected August 5th, 1874, in considerable abundance, on the piles of the wharves at low^ water, — U. S. Fish Commission. 0. hideiitata resembles 0. gelatinosa in the delicacy and grace of its habit, in the flexibility of the compound stem and branches, and in the pellucid whiteness of the upper portion of its branches and branchlets. Campanularia pygmaea, sp- nov. Plate IX, fig. 9. Stem often creeping, with short, stout, coarsely annulated, upright pedicels, sometimes with one or two short, annulated branches, each bearing a single calycle. Ilydrothecpe large, deep campanulate, tapering slightly toward the base, and with longitudinal lines at regular intervals, extending down al)out one-fourth the length of the calycle ; the rim is ornamented with from ten to fourteen square-cut denticles, which are more or less hollowed out above, and separated by rather shallow evenly i-ounded notches, of about the same breadth. Gonotheca? unknown. Height about 1""". Found gi-owing on a specimen of Sertularia latlusoda, from Casco Bay, Maine, — U. S. Fish Commission. 60 S. F. Clark on New and Rare Species of Hydroids. Campanularia noliformis McCrady. Plate X, fig. 5. A few specimens of this species were collected at low-water, near Savin Rock, in the latter part of September, 1 874, attached to Zostera marina. The hydrarium agrees very well with McC-rady's description. We were not fortunate enough to find the gonotheca?. This is the first time this species has been recorded since McCrady described it from the harbor of Charleston, S. Carolina, in 1857. Campanularia calceolifera Hincks. Ann. and Mag. of Nat. Hist., vol. viii, Aug., 1871, page 78, pi. vi. Plate X, figs. 7, 8. Stem filiform, slender, flexuous, sometimes slightly branched, ringed at the base and above the origin of the branches, light horn-coloi', with the upper portions pellucid white; branches short, curving out- ward, undivided, and bearing but two or three calycles. Hydro- thecse alternate, broadly campanulate, deep, with a slightly everted entire rim, and borne on annulated pedicels of variable length, those on the upper portion of the stem consisting of five to eight rings, those near the base, of twelve to twenty. At each bend, of the stem a single hydrotheca is given otf, and tliese all ciirving outward give to this species a very gracefu.1 habit. Gonotheca? axillary, borne on pedicels consisting of three or four rings, largest at the distal ex- tremity and tapering gradually toward the base, with a peculiar in- curved coil or twist at the distal end near the opening ; the apei*ture is shield-shaped and placed in a depression on one side of the distal end. An internal membrane extends inward from the shield-shaped opening and terminates in a circular orifice near the distal extremity. Height about one inch (25"""). Noank, Conn., from the bottom of an old scow, Sept. 9, 1874, with gonothecse; piles of wharves at Woods Hole, Mass., Aixg., 1871, with gonothecse — U. S. Fish Commission. Hincks' figures represent the hydrothecae as being more everted than they are in the American specimens ; otherwise they exactly correspond. This is the first time this species has been recoi'ded from the American coast. «S, F. Clark on Keio ami Bare K<pecies of Ih/drokU. ui Gonothyraea tenuis, sp. nov. Plato X, tig. 8. Stem simple, somewhat iiexuous, slender, and aniiulated ahovc the origin of each of the numerous branches, wliich are an-auged alter- nately, some simple and some compound, the latter l)earing numerous brancldets, the lower branches sometimes half the length of the main stem ; base of the stem and of the lower branches light horn-color, the ui)])er portions of the same and all of the brancldets jjellucid white; branches and branchlets spreading, giving quite a bushy ap- pearance to a Avell develo])ed colony. Hydrotheca' variable in size, deeply campanulate, tapering quite rapidly from a little below the middle to the base ; the rim is ornamented with teeth which show considerable variation, both in number and in shape ; in some cases they are quite sharj) and shallow, while on other calycles upon the same stem they are of a castellated form and sometimes slightly emarginate ; in number they vary from ten to sixteen; the i)edicels which support the hydrotheca? also vary greatly, some being com- posed of but three or four annulations, others of as many as fourteen. Gonothecge axillary, very much elongated, narrow, obconic, taper- ing gradually from the distal to the proximal end, borne on short pedicels of but three to live rings ; the nnmber of medusa? holding planuhe contained in each reproductive capsule varies from two to five and the number of planula^ in each medusa varies to the same extent ; the tentacles of the medusa? vary considerably in length and in number, some of them being over half as long as the diameter of the medusa, Avhile others are not more than one-third that length, in number they vary from eight to fourteen. The planuliB at the time of liberation are regularly cylindrical, and their length is equal to nearly foiir times the width. Height usually 1 to 1-25 inches (25 to 38'""^). New Haven, Conn., on piles of Long Wharf, June 2nd, 1875,— S. F. Clark. Found in consideral)le abundance at low-watei', loaded with reproductive capsules. The large size of the latter, together with the clusters of extracapsular medusa, make this quite a showy species for one of such humble growth, Opercularella pumila, sp. nov. Plate IX, figs. 3, 4, 5. Stem rather stout, erect or creeping, slightly flexuous, amudated throughout, sparingly branched ; branches erect, undivided, some- 62 S. F. Clark on New and Rare Species of Hydroids. times attaining a considerable length. Hydrothecae largest in the middle, tapering very slightly toward the base, rapidly converging at the distal end, and supported on short annulated pedicels, consisting of three to five rings. Gonothecje fusiform, with the tapering neck often somewhat elongated ; length about twice that of the hydro- thecffi. They contain one to three small globular or ovate immature raedusoids. The pedicels consist of three to six annulations. In the creeping form the hydrothecre appear at intervals borne on short ringed stalks consisting of about three to six rings. Portland, Maine, August, 1873, with gonothecae, on piles of wharves; off Montauk Pt., Long Island, 5-15 fathoms, August, 1874, — U. S. Fish Commission. This species closely resembles O. lacerata of Hincks, from which it is distinguished by the forms both of the hydrotheca? and gonothecae. The hydrothecae are also smaller than those of 0. lacerata. As the reproduction has not been traced in this species, it can only be refer- red to the genus provisionally. I am inclined to think, from the shape of the gonothecte and from the fact that they often contain two or three distinct reproductive bodies, that it may not belong under Opercularella. Opercularella lacerata Hincks. Campanularia lacerata Johnston, Brit. Zooph., p. Ill, PI. xsviii, fig. 3. Opercularella lacerata Hincks, Brit. Hydr. Zoopli., p. 194, PI. xxxix, fig. I. Plate IX, fig. 6. Stem erect, simple, slightly flexuous, more or less annulated through- out, sparingly branched ; branches short undivided. Hydrothecae ovato-fusiform and borne on short pedicels of but two or three annula- tions; operculum composed of six to eight segments. Gonothecae, of the female colonies, a trifle wider at the distal end, and tapering very slightly toward the base, supported on short ringed pedicels ; the medusoid (sporosac) containing the planuhi? is quite large, the diameter being about equal to the length of the gonotheca ; from two to five planulfe in each medusoid. New Haven, Conn., on piles of Long Wharf, May 13th, 1875, with extracapsular medustB. — S. F. Clark. The hydrothecae in this species average about one tliird larger than those of 0. pumila ; the segments of the opercula are more deeply cleft ; and there are differences in the form of the gonothecae. *S'. F. Clark on Xew and Hare Species of JlydroiiU. 63 Calycella syringa "incks. Peculiar variety. Plate X, figs. 1 , 2, 3. Sertularia syringa L'mn., Syst. 1311. Calicella syringa Hiucks, Oat. Devon Zooph., 23; Aim. N. 11. (3d .son), VIIT, 294. Calycella syringa Hiueks, Jiritisli Hydroid Zoophytes, Vol. I, p. 206, Plate xxxix, figs. 2, 2a. Stem simple, creeping, nearly smooth. HydrothecjB hyaline, color- less or tinged with a light horn-color, cylindrical, romided oft" helow, with an everted rim, to which is attached an opercnlnm consisting of from five to eight segments and supported on twisted j)edicels of considerable length, with eight to fifteen twists; some of the liydi'o- theca' have an addition in the shape of a wide ring, ornamented with from ten to fourteen longitudinal markings, which rises for some dis- tance above the rim and on the summit of which there is borne either the operculum or another ring ; in some cases there are as many as four of these rings with an operculum at the summit. The opercula usually point upward, but are occasionally deflected into the ealycle. From Casco Bay, Me., 9 fathoms, — U. S. Fish Commission. Halecium articulosum, sp- nov. Plate X, fig. 6. Stem dark brown and tapering gradually, very stout, sparingly branched, compound, consisting of many, slender, anastomosing, ser- pentine tubes ; branches short and irregularly arranged on all sides of the main stem ; branchlets few and very short ; both branches and branchlets are divided into very short, stout internodes by distinct joints placed at right angles to the stem ; branches and branchlets simple, whitish, delicate, becoming more numerous toward the top of the stem. The internodes become shorter very gradually toward the ends of the branches and branchlets. Hydrothecte alternate, short and wide, one to each segment ; some of them have a cup within a cup, as is so often the case in the species of Halecium. Gonothecje borne in rows on the upper side of the pinnae. The female gono- thecaj are large, obovate, and have the opening on one side and nearer the distal than the proximal end ; the male gonothecse are oblong, subcylindrical, and, like the female, are sessile. Height of the largest specimens, 5 niches (125'""'). Eastern end of Long Island Sound, 8-12 fathoms; Coxe's Ledge, S. E. of Block Island, 17-21 fathoms; Casco Bay, Maine; Eastport, Me., — U. S. Fish Commission. 64 S. v. Clark on New and Rare, Species of Ilydroids. H. articulosiini resembles H. pluniosiini^ but has a stouter habit ; the iuteruocles are shorter and stouter ; and it may also be dis- tinguished by the direction of the joints, at right angles to the stem, and by the very wide-mouthed calycles. The female gonothecse some- what resemble those of M. Beanii^ but are of a stouter build ; they are relatively larger at the distal extremity, the orifice is differently shaped and is differently situated, being nearer the distal extremity. Sertularia argentea Ellis and Soiander, vaj\ divaricata nov. Plate X, fig. 7. Stem simple, stout, erect, straight or slightly flexuous, of a deep horn color, regularly jointed, each joint having two or three branches ; branches alternate, sparingly branched, diverging at right angles from the main stem and all in the same plane, divided quite regularly by joints, each bearuig two pairs of hydrothecte, much resembling a young shoot of the usual form of S. argentea. Hydrothecte nearly opposite, curving strongly outwards, with a bilabiate mouth, the upper lip being considerably smaller than the lower ; hydrothecse are also scattered along the stem in pairs. Gonothecae unknown. Collected at Oasco Bay, Me., 1878, — U. S. Fish Commission. I at first thought that this was a distinct species, but I have since had intermediate forms which prove quite conclusively that it is only a variety of S. argentea. Considerable variation is shown in the hydrotliecae of this variety ; some of them, on the same stem, are more directly opposite and curve outwards more than others. Plumularia Verrillii, sp. nov. Plate X, fig. 9. Stems erect, simple, straight or slightly curved, slender, two to four inclies high, of a bright horn-color, branched and regularly jointed by transverse divisions; the branches have their origin near the base of the stem, are ei-ect and resemble the main stem in all par- ticulars; pinnae occasionally branched, regularly arranged on two sides of the main stem and branches, sej^arated by an angle of ninety degrees, composed of long similar joints, each bearing a hydrotheca and a number of nematophores ; occasionally there is an odd, intermedi- ate joint bearing only one or two nematophores and no hydrotliecae ; a single pinna to each joint. Nematophores sessile, compound, large, tapering to the base, with a round cup-like opening : there are four to six on each hydrotheca-bearing joint, one on eacli side of the upper S. K Clark on JVew mul Jiare Species of TTydroida. f55 edge of the hydrotheca, two or three between the Iiyth-otheca and the proximal end of the joint, and occasionally one at the distal end ; on tlie main stem there are usnally two in each axil and two or three on eaeli joint. ITydrotlieca^ small, sliallow, ahout equal in depth to the length of the nematophores, attached to the stem by their entire length; rim entire. Gonotheca? borne in the axils of the stem and of the branclied ])innfe, sessile, tapering at the base, the remaining portion either i-egularly cylindrical or slightly sw^ollen in the middle; aperture large, terminal. Height, 2-5 inches (64"""). Eastport, Maine, 10-20 fathoms, 1868,— A. E. Verrill and S. I. Smith. P. Verrillil is a beautiful little, peHucid, white species, with a deli- cate, graceful habit which readily distinguishes it from any of the forms now known upon our coast. It is the second genuine Plumu- laria from the New England coast, both having been discovered by Professor Verrill. The previously described species {P. tenella Ver- rill) was dredged in 1871, off Gay Head, Martha's Vineyard, in 10 fathoms. It has since been dredged in 4-5 fathoms oif the Thimble Islands, near New Haven, Conn., and it was also found on the piles at Greenport, Long Island, August 5th, 1874, with gonothecae. It differs greatly from the present species in the form of the gonothecae, w^hich are in the shape of an elegantly curved cornucopia, slender at the base and gradually enlarged to the end, and with a cluster of nematophores at the base. The hydrotbecae are also different in form. EXPLANATION OF PLATES. Plate IX. Figure 1. Ohelia hicuspidata ; from Thimble Islands. Figure 2. Ohelia bidentata ; from Greenport, Long Island. Figure 3. Opercularelld pumila ; creeping form. Figure 4. The same, yoimg, with stem erect ; from ofiE Montauk. Figure 5. The same, from Portland, Me., showing a more luxuriant growth ; a and c, the hydrarium ; h, gonotheca, enlarged 32 diameters. Figure 6. Opercularella lacerata ; a, hydrarium ; b, gonotheca ; c, medusoid ; d, un- developed planulse. Figure 7. Campanularia calceolifera ; from Noank, Conn. Figure 8. The same ; a, hydrotheca ; b, gonotheca ; from Noank, Conn. Figure 9. Campanularia pygmcea ; from Casco Bay. Trans. Conn. Acad., Vol. III. 9 July, 1875. 66 )S. I^. (J lark on New and Rare ISpedes of Hydroids. Plate X. Figure 1. Galycella syringa ; from Casco Bay, showing a peculiar variation in the operculum. Figure 2. The same, showing the variation in the size and shape of the hydrotheeas and in the length of the pedicel. Figure 3. The same, with one secondary ring. Figure 4. Clytia Johnstoni, from Noank, Conn. Figure 5. Campanularia noliformis, showing variations in the pedicels. Figure 6. Halecium articulosum ; from Coxe's Ledge. A, a branch bearing both hydrothecte and gonothecse ; a, gonothecas ; 6, hydrotheca3. B, a branch with hydro- thecae only ; h, a branchlet ; c, hydrothecis. Figure 7. Sertularia argentea, var. diva?-icata ; from Casco Bay. Figure 8. Gonothyrcea tenuis ; a, branch with hydrothecae ; h, gonotheca with extra- capsular medusa; ; c, medusa with radiating tubes and tentacles ; d, planulse. Figure 9. Plumularia Verrillii; a, branch showing hj^drotheca; and nematophores and the arrangement of the joints ; b, gonotheca ; c, a single joint. III. Ox TIIK ClIONDUODITE FROM Till': Tll,I,Y-F()STKK IkOX MlNK, BuKWSTKR, Kkw York. By Edward S. Dana. WITH THREE PLATES. The interesting discovery by Seacchi,* of the existence of three types in the crystals of the Yesnvian huniite, gives especial interest to the study of chondrodite — a mineral identical with humite in chemi- cal composition, and yet very diiferent in appearance, as well as in origin and method of occurrence. The same subject of humite has since been more exhaustively investigated by vom Kath,f with the entire confirmation of Scacchi's views. These authors have shown that the crystals of humite are to be divided into three groups, all bear- ing the same relation to each other in respect to their lateral axes, while the vertical axis has a distinct value for each type. In other words, the planes occurring upon a given crystal bear simple relations to each other, whereas only very complex symbols result when the planes of one type are referred to the axes of another. For a full explanation of this subject reference must be made to the valuable memoirs al)ove alluded to. It will be sufficient to give as an example the symbols of the occurring pyramids of the r series on the second and third types of chondrodite (see beyond) ; (1) as referred to their own axes ; (2) as referred to the axes of the second type. (1 II. •) III. II. (2.) III. J-2(r') ^-2(P«) 4 - 36 _ 55-2 i2(.^) 1-2 ip^) 4 - --■> 36 _ 45-2 \Mr^) \-2{p') 4 _ — •) 3 - 36 _ 35 ' 4-2 (H) Ui,^) 4-2 36 . 25-2 \-W) 36 _ r5-2 8-2 (p8) 36 _ * rogg. Ann.. Erg. Bd. iii, 161, 1851. f Pogg. Ann., Erg. Bd. v, 321, 1871 ; vi, 385, 1873. 68 E. S. Dana — Chondrodite from the Tilly- Foster Iron Mine. The chemical composition of the three types of humite has been most recently investigated by vom Rath,* and although analyses lead to somewhat different results in the three cases, he concludes that in composition they are still essentially the same, and that the cause of the variation in crystalline form is not to be found in the relative amount of fluorine present, as has been often assumed. A further remarkable peculiarity true of two of the three types is their hemihcdral character, which is clearly set forth in the memoirs referred to. These points are alluded to here because of their direct bearing on the crystallization of chondrodite, which forms the sub- ject of this j^aper, Chondrodite was first shown by Rammelsberg to be identical with humite in chemical composition, but its ci-ystallographic relation to it was not brought out until the investigations of Kokscharow. He showed, in his " Materialien zur Mineralogie Russlands," vol. vi, p. 73, 1870, that the crystals from Pargas, Finland, were identical in form and angles with type II of humite. Vom Rath has followed with the description of crystals from Nya-Kopparberg, Sweden, and px'oved that the same fact is true of them. The study of the chondrodite from the Tilly-Foster iron mine, Brewster, Putnam Co., New York, which I have been able to make during the past season, has shown that it, too, is for the most part iden- tical in crystalline form with type II of humite, but that at the same time crystals exist belonging to type I, and others which belong to type III. Further than this, the chemical composition of the second type crystals, as shown by an analysis by Mr. G. W. Hawes (p. 21), agrees with great exactness with that of the Swedish mineral anal- yzed by vom Rath. Moreover, the detailed study of these crystals has shown that while they agree with humite in the character of their hemihedrism, as well as in angles, they surpass it in the multi- plicity of secondary planes. Thus a single solid angle has been observed M^hich was modified by fifteen distinct and well-defined, though very minute, planes. This, as will be seen when the facts are described in detail, implies a delicacy in the action of the crystallogenic forces at work which is unparalleled, and sustains the opinion that chondrodite, or humite, is unique among mineral species. The method of occurrence at the Tilly-Foster iron mine has been fully described by Prof. Dana in a memoir entitled, " Serpentine pseudomorphs and other kinds, etc.," Journal of Science, viii, pp. 371, * Pogg. Ann., cxlvii, 246, 18'72. E. S. Dana — Chondrodite from the Tilly-Foster Iroti 3Ime. 69 447, 1874. It may be of interest, liowever, to review tlie subject again so far as the chondroilite itself is immediately concerned. The chondrodite forms the gangue of the magnetite, being every- where disseminated through it in varying [troportions. In tlie parts of the mine wliere the ore is purest and perfectly firm and solid, — the so-called " blue ore," — the associated chondrodite is sparsely sprin- kled through it in small yellow grains, showing no trace of crystal- line form. Occasionally, however, the firmer ore contains the chon- drodite in very large but im})erfect crystals, or crystalline masses, associated directly with enstatite and enveloped with dolomite, which have a dark, rich brown color, and a brilliant luster on the fracture. A distinctly laminated structure is uniformly ])resent, which is per- haps due to cleavage (?) (See page 21.) Isolated grains imbedded in dolomite often show traces of crystalline faces, though nothing that admits of even approximate determination. An analysis of this vari- ety of the mineral gave Mr. Breidenbaugh (Am. J. Sci., Ill, vi, 209), Si 35-42, Fe 5-72, Mg 54-22, Fl 9-00 = 104-3G; equivalent of oxygen replaced by- fluorine, 3-79. In the lai'ger portion of the mine as no-w opened the soft " yellow ore" predominates : the chondrodite is present in it in much larger quantities, and, like the other minerals present, it has almost uni- versally suiFered extensive alteration. A long list of these products of alteration has been fully described by Prof. Dana in the memoir already alluded to. The chondrodite forms the main portion of the material taken out, and many tons of this refuse matter are yearly thrown away. It vai'ies much in color, but is generally of a light yellow; it iisually has more or less of a soapy feel and shows a vari- ety of transition-products between the semi-altered material and serpentine. The chondrodite in this " yellow ore" is generally mas- sive ; but occasionally fragments of large coarse crystals have been found, some of which measure five or six inches in length. These are always more or less altered ; moreover, the material of which they are formed is far from homogeneous, masses of magnetite, and also chlorite, being often enclosed. Dolomite is the most constantly associated mineral and occurs in rhombohedrons of considerable size ; these, as well as the crystals of chondrodite, are often coated ■with magnetite. Better crystals of chondrodite than those just mentioned are some- times found in what Avere once cavities in this massive material. Unfortunately these have all suffered from the general alteration and now have little or no luster, and often are not even smooth. These cavities are almost invariably filled with a soft mealy serpentine, 70 E. S. Dana — Chondrodite from the Tilly-Foster Iron Mine. which can be cut out with a knife. These crystals vary in size, being sometimes an inch or two in length. A crystallographic examination of them is seldom possible, but a few of the crystals found allow of it, and the results are described beyond. The form is usually very simple, and the color varies from a deep red to a light yellow. This may be said to be the common method of occurrence at the locality. Forti;nately, matei'ial much l)etter adapted for crystallographic study also occurs, though this is very rarely true. Narrow veins are sometimes met with, two or three inches across, which were originally lined with more or less perfectly crystallized chondrodite and also with dodecahedrons of magnetite, crystals of rijDidolite, and rarely apatite, and then subsequently filled in with dolomite. Where this has been the case and the dolomite has remained intact the chondrodite has been protected and the crystals have retained per- fectly their luster and color. Only in a few instances were the ciys- tals polished when covered simply by a soft serpentine. The chem- ical composition of this chondrodite is given beyond, after the description of the crystals. It has a deep, gai'net-red color, and a luster equal to that of the finest Binnenthal blende. 1. Description of Crystals belonging to Type II. The remark of vom Rath in regard to the irregularity of form of the Swedish chondrodite is eminently true of the Bi'ewster crystals. For in the same little group no two are alike ; so that each one de- serves and requires an especial study. The first point to be determined was the values of the fundamental angles. Some difficulty was found in obtaining these from the fact that many crystals, though faultless in luster, yet gave uncertain measurements. This was due to the fractured condition of many of the planes, which, though often not very apparent at first sight, yet gave rise to a variety of reflected images in the goniometer, of which no one could be accepted as trustworthy. All the larger crystals show a multitude of internal fractures; and, where such crystals have been subjected to altering influences, this circumstance has has- tened their destruction, and in all cases the external condition of the planes has been more or less aflected. The direction of the fracture lines was in most cases entirely irregular, though in a number of cases they were distinctly parallel to e^{-l-i). The presence of these cracks gave the crystals the appearance of having suffered sudden contraction, by which the planes had been irregularly drawn inward, forming re-entrant angles ; in fact, in this respect, as in general appearance, this chondrodite might be aptly compared to a resin. E. S. Dana — C/iondroilitefrom the TiUy-Foster Iron Mine. 71 Tlie smallest crystals proved to be free from this cause of irregu- larity, and one of tliem, on which the faces were exceptionably bril- liant, was chosen for careful measurement. It may be added that all the nieasui-cnients were made Avith an Oertling goniometer, pi-ovidod with two telescopes. The mean of 30 measurements of A {0=.00\) on r' (—-2^247) gave: 135° 18' 50". The maxinumi variation from the mean given was ±45". The mean of 30 measurements of ^ on (i"|— -^=z205 J gave: 140° 55' 48". Maximum variation -4-45". These were accepted as the fundamental angles, and as the agree- ment l)etween the other anoles measured and the calculations made Table I. Ghondrodite. Hwmite. (7 = ^(010) Calculated. A=O(001) Measured. Calculated. A (v. Rath) Calculated. i l-l Oil 147° 32' 39" 122° 44' (ap.) 122° 27' 21" 122° 27' 49" ea 2„ 5" 205 90 *149 55 48 149 58 48 e' \^ 203 90 135 59 136 1 17 135 52 15 c'^ 2-1 201 90 109 4 109 3 24 108 57 50 r> 4- 247 129 42 9 *135 18 50 135 18 50 135 17 40 r« 4- 245 137 25 45 125 52 125 50 6 125 49 0 ^ 4- 3^ 243 146 27 42 113 25i 113 25 36 113 24 45 r* 4-2 241 154 2 9 98 14 98 13 6 98 12 47 m? -5 641 125 43 56 95 22 95 19 40 95 17 59 w> 2 3 223 127 1 31 125 3 49 125 2 47 «« 2 221 135 45 24 103 11 103 10 4 103 9 35 upon the above basis proved to be as close as could be desired, hav- ing regard to the nature of the plane in each individual case, no 12 MS. Dana — Chondrodite from the Tilhj-Foster Iron Mine. attempt to correct them was deemed desirable. Calculated from these angles the parameters are : a (vert.)=l-57236 ; ^'^l; c=l-08630; and the angle for the fundamental prism is J/s,Z(llO/sllO)==85° 15' 46" or 94° 44' 14". It may be added that the angle of A on / behind (siY) was meas- ured with equal care, and found to be 135° 18' 40". The preceding table* (I) includes the principal angles measured on the same crystal, and also those calculated from the above parame- ters ; in addition, the corresponding angles for humite, type II, are also given, as calculated by vom Rath. The angles of the maci-o- domes agree very closely, it will be observed, in chondrodite and humite ; in the brachydomes, on the other hand, there is a divergence of 6 or 7 minutes. The angles given in table I, and also in tables XII, XIII, XIV, for types I and III, are the actual angles. In all the other tables, however, the supplement (normal) angles, as measured and calculated, are uniformly given. The reason for this was that these angles hav- ing special reference to the sphere of projection, and being chiefly of value in calculating with it, it did not seem worth while to change them from the form in which they had been used. It is necessary to explain, also, the system of symbols and let- ters here and subsequently employed. The fundamental foi-m adopted is the same as that used in Dana's " Mineralogy," and first suggested by the author of that work in Am. J. Sci. II, xiv, 175, 185. It is to be remembered that Scacchi made the prism of the r series (i. e., «-3) the fundamental prism, and gave to the vertical axes lengths, in the ratio of 7 : 5 : 9 to each other, for types I, II and III respectively. Vom Rath followed him in this latter respect, but for the vertical prism took that of the 7i series (i. e., 7, or 110 of Dana). Prof Dana took a modified view of the relations of the three forms, and chose for the fundamental macrodome in each type the plane making with A{ 0) an angle of 122° to 125° : thus on humite (vom Rath), type I, 124° 17', II, 122° 28', III, 125° 15'. In this view the vertical axes have one-third (I), one-half (TI) and one-quarter (III) of the lengths assumed by Scacchi ; in other words, their relation to each other are * Both the symbols of Naiimann (in the form used in Dana's "Mineralogy") and also of Miller are given; the signs belonging to each plane omitted Jiere, as the relations of the planes are shown with sufficient clearness on the spherical projection, Plate xiii. E. S. Dana— Choiidrodite from the IWij- Foster Iron Mhic. 7.3 as 7 : |: fl or 28 : ;^0 : 27. Tlie method adopted by Seacclii and vom liatli lias the advaiitao-o of exi)ressing the sinij>lest })Ossil)le iminer- ical ratios between tlie three tyi)es. It is the view of tlie aiitlior above referred to, however, tliat the variation in the anuU' ot ^^1-7(01]) is to be regarded in tlie same light as the variation in the vertical axes of the rhoiubohedral carbonates, or in the orthorhombic sulphates ; or in other Avords, the three types form an isoniorjihous series, and the variation observed is no greater than is constantly seen in analogous isomorphous groups. This view seems to find con- firmation in the crystallographic relations of humite and chrysolite, a subject already discussed by Scacchi, Rammelsberg and vom Kath, Taking the fundamental form, as in Dana's " Mineralogy" (here i-sz^T" of other authors), the lateral axes are nearly identical with those of humite, while the vertical axis (1 •25928 Kokscharow) has exactly the ratio of -^jj to that of humite, type II, and f to that of humite, type III ; in other words, we have the ratios : Humite. Chrysolite. III. I. II. 24 : 27 : 28 : 30. If we adopt the vertical axes of Scacchi and quadruple that of chry- solite, we obtain II. I. Chrysolite. III. 5:7:8:9 These relations were in effect lirought out by Scacchi when he showed that what he called the common fundamental form of humite, ob- tained by dividing the vertical axes by 7 (I), 5 (II), and 9 (III) re- spectively, was nearly identical with that accepted by him for chry- solite. This fact seemed to Rammelsberg of so much importance that he proposed to refer all the planes of humite to this common funda- mental form ; and in this he has been folloAved by Kokscharow. The result of this will be seen in the following table, which gives the symbols thus obtained for the two common macrodomes in each tyj)e. I. II. III. ro m 5/1 \ §(2'7 m I 0-^) 1 (!-') 9 8 ('-*' I i^-') -: (-) Tr.\ns. Conn. Acad., Vol. III. 10 July, 1875. H E. 8. Dana — (Jhoiidrodite from the Tilly-Foster Iron Mine. It will be seen from these few examples that the plan proposed in- troduces a set of common molecular axes at the expense of all crys- tallographic simplicity in the relations of the several planes of each type. Moreover, the view of Rammelsberg loses some of its plausi- bility, if, as shown by vom Rath, the vertical axes do not stand in direct relation to the amount of fluorine present. The view of Prof. Dana here advocated seems to have the advantage of presenting all the relations in their most natural light. It may be added, as completing the history of the subject, that DesCloizeanx refers all the humite planes to one, and that the second, type. In regard to the letters employed, it seemed to ofler the simplest solution of an obvious difficulty to retain all the letters of Scacchi for the second type, and for the third to use simply the corresponding Greek letters in the same order, and for the first type to use the cor- responding capital letters. It was not deemed advisable to use the same letter for two planes, on different types, which bore no imme- diate relation to each other. Table II. A= O(OOl). Table III. C= w(OlO). Calcu- lated. II. III. IV. i' \-i Oil 57° 33' 57° 28' ea h 205 30 4 el h 203 43 58f e2 2-1 201 70 57 yi 4 _ 247 44 41 j44 404 ] 44 40 44° 37' 44° 41' r" 4 _ 245 54 10 54 12 y3 4 _ 3-2 243 66 34 i 66 39 ( 66 32 66 25 66 35 ^ 4-2 241 81 47 81 52 «,! 2 3 223 54 56 «,2 2 221 76 50 rn^ el 2 641 84 40 Calcu- lated. II. III. IV. 32° 27' i 32° '( 32 28' 32 90 90 90 50 18 j 50 1 50 7 30 ( 50° \ 50 13' 30 50° 12' 42 34 42 39 42 32 42 35 33 32 j 33 } 33 31 32 33 30 33 27 25 58 1 26 ] 25 0 57i 44 14J 44 11 52 58 54 16 54 18 E. aS. Dana — Chondrodlte from the Tilly- Foster Iron Mine. 75 In the tables, II, III, IV, are given the angles as measured in a series of crystals (each crystal is numbered). They are import- ant as showing how far the angles are constant. Some considerable variations from calculated angles in a few instances are to be ex- plained by the cause of irregulai'ity already mentioned — the occur- rence of irregular fractures across the planes. In table V are given the su])plement angles calculated for all the planes on e- and also the angles measured on the several crystals (as before numbered). The angles ai-e in pairs corresponding to 201 and iiOl, or 3OI and 201 in each case. (Compare the spherical projection, plate xiii.) Table IV. Angles measured on C = i-i (010). V. VI. VII. VIII. IX. X. XI. XII. r\ 4 _ 247 50° 11' ( 50° 16' ] 50 16 50° 50 17' 17 50° 12' 50° 24' r^ 4 _ 245 42" 34' (42° (42 20' 12 (42 30 ]42 23 42 28 j42 142 36 29 42° 34< ( 42 38 ]42 36 r3 4 _ 3-2 243 33 26 33 32i j 33 ]33 33 34 33 27 33 32 r* 4-2 241 25 59 (20 V26 1 0 25 55 25 57 25 56i 26 0 n^ 2 3 223 (44 ] 44 20 10 (44 ]44 26 14 44 14 44 15 n- 2 221 52 57 52 55 52 52 m" 3 «-2 641 54 31 Of the planes which occur, according to Scacchi and vom Rath, on type II of humite, I have identified all but m of Scacchi (f-|) and f-e of vom Rath (f-t). Of new planes I have found the following, which fall in the old vertical or horizontal zones, and many others to be described later; o (;-2'=2]0), r (2-7=021), ia (*-i=047), //3=(|-I= 025), ea(f-?=205), r«(|-5=489) ; of these the most interesting is the prism i-2, as hitherto no prism has been found on either the 2d or 3d types. Hemihedrism — The ])eculiar hemihedral character of humite has been alluded to, and it is a little striking that the crystals of chon- drodlte should show so entire a corres])ondence to it. Taking the same position for the crystals as vom Rath, r- and r^ appear uni- formly in the positive (or upper) quadrants, r' and r^ always in the 76 E. S. Dana — Chondrodite from the Tilly-Foster Iron Mine. Table V. Angles measured on e- = 2-j (201 and 201). e' 2 203 ri 4 - 247 ri 4 - 245 y3 4- 243 ^ 4-2 241 n' 2 3 223 n^ 2 221 m 641 Calculated. 26° 57' 53' S 59 19 58 ■( 87 22 38 j 59 14 0 ( 82 34 28 60 30 76 32 0 14 j 64 2 9 { 69 51 3 j 44 38 20 { 70 20 31 j 45 45 24 / 56 42 18 37 34 34 42 57 12 26° 55' 59 14 44 34 45 40 45 50 YII. 59° 33' (approx) 76 26 64 0 45 46 56 42 IX. 59° 30' 59 30 76 15 63 57 56 26 42 43 42 32 26° 51 59 20 82 34 45 42 56 42 42 51 59° 20' 76 35 VI. 26° 59' XI. 64° 0' negative (or lower) and 7\^ is both + and — , but where occurring alone is generally negative; n'^ is generally, and m^ always, nega- tive. Of the brachydomes it may be said that they are often holohe- dral, but this is not always the case. The various figures on the two plates will show the ti-ue relations better than words. It is to be said, however, that when the brachydomes are ± they are still distinguished from each other physically. Thus the -f- series may be largely devel- oped and rough, destitute of any semblance to polish, when the negative series is as lustrous as the pyramidal planes. When e^ (2-?=:20l) is only once present it is uniformly positive. The macrodomes are always holohedral on humite ; here this is sometimes the case, but there is a good deal of irregularity (as will be seen in the figures), and this is conspicuous in figiire V where i and ik occur together and also e«, ?/?, and i. Hahit. — With regard to the general habit of the crystals, it is inter- esting to note the wide variation which is shown. Figui-es 1, 2, 0, 9, 10, are intended to give some idea of the crystals as drawn symmet- rically, and figures 7, 10, 14, 15, 16, 17, 18, 19, , of their actual a2:)pearance. As will be seen, the figures are drawn with C(^-^, 010) in front : this was necessary in order to give a true idea of their real appearance. The prism ^-3 is so acute (49^°) that when directed E. S. Dana — Choudrodite from the T'llli/- Foster Iron Jfi/ie. 77 toward the eye the projection gives it but little width. As a mat- ter of fact the crystals have C uniformly well developed and are generally attached approximately by the extremity of the bracliy- diagonal axis. This having been explained, it will be clear that wliilc tig. 3 is an almost exact rei)roduction of an actual crystal, Hg. 5, by the other method of projection, gives an entirely false idea of its ap])carance. It is certainly true that the latter method shows the hemihedrism in its true light, but this should not weigh against the other more important consideration. The crystals from Avhich the partial figures, 7, 10, 16, 17, were drawn were united, along with others quite as diverse, in one small fragment only half an inch in length. It is to be noted that figures 14 and 15 are really more different than would appear at first glance. The crystals drawn in figures 16 and 19 also occurred closely conjoined in the same group ; and other examples might be mentioned. One crystal of a very prismatic appearance (when placed in an inverted position) is shown in figure 1 9. Presence of minute />/«>* ^^s. — The most remarkable feature of the mineral from this locality is yet to be mentioned. I refer to the mul- titude of minute planes which modify many of the solid angles. One single case will be discussed in detail, as the planes admitted of more than usually exact determination ; it serves well to illustrate the subject. A horizontal projection of a portion of the crystal is shown in fig. 14. The crystal itself was small, and unfortunately so imbedded in dolomite that it was for the most part rough and be- yond even approximate measurements. The part available shoAved G (i-1) faultless; also r^ good; and less satisfactory r^, r^ and r*. On the solid angle between 6', r^ and r^ a large number of miniite planes were observed ; they were so extremely small (all covering a surface not "OS of an inch in breadth) that any exact measurements seemed at first hopeless. They were sharply defined, however, and brilliant, and when the attempt was made it was found that they gave perfectly distinct though fiiint reflections. It may be remarked here that measurements in this case were only made possi- ble by the substitution of a cross, cut in tin foil and illuminated very brilliantly by a gas l)urner, for the ordinary spider lines in the second telescope — a device for Avhich I am indebted to Prof Schrauf of Vienna.* The measurements were all made with the greatest care, though, as will be readily ixnderstood, the exact adjustment of planes so small * Ber. Ak. Wien, Ixvii, 1873. 78 E. S. Dana — CJiondrodite fro^n the Tilly -Foster Iron Mine. was not an easy task. The symbols were calculated from the angles thus obtained. But as even then some doubt remained as to the de- gree of dependence which was to be placed in them, the measure- ments were all repeated with the same care as before. Tlie result was perfectly satisfactory, as the variation in no case exceeded tlie probable error of observation given to each angle when measured for the first time. This variation in most cases did not exceed ±2'. The following supplement angles were obtained for the more promi- nent planes: C /^r'^ gave 33° 33' (required 33° 32f ) ; and C/^r* gave 25° 53' (required 25° 58'); and r^^r^ gave 31° 31' (required 31° 38'). The following table contains the angles for each of these mi- nute planes as measured on 6', and r"^ and also on ^-, itself one of this group. Table VI. G=i-l{010). r3 = 243 . 1^=201. Meas. Calc. Meas. Calc. Meas. Calc. 1 i- 2-1 021 17° 37' 17° 38i 23° 58' 23° 49' + a;i T- 2-12-7 22 9 21 53 31 4 31 12 8° 21' 8° 37' + a;2 r- 2-26-9 12 55 12 58 29 32 29 27 6 36 1 6 35 + a;^ r- 2-26-7 10 34 10 30 30 37 30 42 8 50 8 54 + x^ 34 It T Y 4-34-'7 9 42 9 39 33 41 33 41 12 0 11 53 -a;" ¥-- I-24-14 20 21 20 28 21 11 21 6 3 25 3 24 -a;5 13 13 12 T 5-13-12 34 32 34 27 7 32 7 15 21 15 21 8 -a;' ¥- ri3-7 18 50 19 17 19 40 19 47 4 5 4 3 -«« ^:- 6-24-13 22 24 22 31 12 15 11 59 12 22 12 22 -x^ 5 15 2" Y 2-15-6 16 5 15 47 18 51 19 7 7 39 7 34 -a;'» 13 13 7"T 4-13-7 24 22 23 59 9 26 9 47 15 25 15 3 -v"- 25 25 7 -25 -9 ]8 54 19 2 14 45 14 38 14 40 14 52 -t -1 9-24-8 21 55 22 4 13 18 13 14 19 32 19 29 -t -1 291 11 43 12 13 23 31 23 14 -y3 -¥ 9-24-1 18 45 19 5 25 34 25 27 K. S. Dana — ChondrocUte from the Tilltj- Foster Iron Mine. 79 The calculated symbols are also striven with the aii^jles which be- long to theui. It will be noticed tliat /- (27 = 021) is itself one ot the minute planes of the same character as those suironndino- it, and its presence gives a reality to them whiclj they would not otherwise have, and .shows what degree of reliance is to be placed on the angles. Cases of a similar character will be noted hereafter. The symbols * calculated for this series of ])lanes are certainly not simple; and yet a moment's consideration will sliow that this was exactly what Avas to be expected. Crowded togetlicr so closely, they would be abnormal if occurring on crystals of any species, while this becomes still more true for a mineral like chondrodite. The constantly recurring, common planes have ratios which in anv other species Avould be considered next to impossible : thus, in tyj)e II, 1 : -^ : -i-: I ; and in type III, 1, ^, ^, 4, ^, ^^. It is to be noticed that these are the true ratios of the r series of pyramids, which exist no matter what change be n\ade in the assumed axes. It is not sur- prising, then, that these secondary planes should themselves have sym- bols totally at variance with the accepted law of simplicity of the indices. Many cases of planes with what may be called abnormal indices have been described, but frequently they are to be explained as has been done by Brezina in the case of the f|--7 (25-0'24), which he proved to exist in wiserine, as a tendency to a plane Avith a simjjle index (1-7=101), which has resulted in a plane Avith approximately the given index. The case in hand, hoAvever, is quite different. It will be noticed, hoAvever, that, laAvless as they appear at first, there is an attempt at system in the symbols given. Thus in the ratio of the brachydiagonal to the vertical axis, we have : x' 12: V x' 13 ; ; 7 x' 12 : 1 x'" 13; ; 7 y' 24: 13 x' 26: 7 t 24: 8 x' 26: 9 y' 24: 1 x' 13: 12 and so on. The ratios for the other axes might be draAs^n out in the same Avay, but they are already contained in the symbols given in the table. A little surprise is felt at first that the uniformity in ratio is not greater, that, for instance, x'^ is not 2-13*7, instead of 2-12"V; but the measurements are too good and reliable to allow of such a * In the symbols given in the tables (i. e., those according to Naumann) the dash over the second figure, or fraction, has been omitted (in order to simpHfy the work of the printer). This has also been done in all the following tables, being made possible by the fact that all these planes, with one or two exceptions, belong to the macrodiagonal series. 80 E. K Dana — Chondrodite from the Tilly-Foster Iron Mine. supposition. It is to be noticed that when ratios of this character are allowed, a slight change in the measured angle will alter entirely the calculated index ; the liberty in this respect is not, however, quite so great as it would stem at first sight to be. For example, the ratios t(M) ^"<^ f(f f ) approach pretty closely to each other, and it might be a question which was to be accepted as the true ratio of the two axes for a certain plane ; and yet if the ratio of one of these axes with the third be unqiiestionably expressed in sevenths, e. g., f , then there seems little doubt but that the ratio f is to be accepted, for that would give 8*4'7 or f-2, while the other supposition would give .35-72'63 or|--^f. This principle has been accepted in obtaining all the indices given in the following tables. A remarkable fact connected with these planes, — in fact, implied in what has already been said, — is that there is so little tendency among them to lie in zo^es. For example, cc^, a-^, y~ and y^ lie very nearly in a zone with each other and Z^, and yet the reflections in the gonio- meter deny that this is exactly true, while no satisfactory indices can be obtained on this supposition, (.e^, i^ and y^ are, however, in a zone.) In regard to these planes two points are to be noticed. In the first place, the question suggests itself whether, if referred to a common fundamental form (see above), or to that of either of the other types, the relations of the planes would be at all more simple. This is an- swered in the negative, as will be seen to be necessary if the trial is made, and also evidently because planes whose normals make angles of a few degrees only with one another can never bear simple rela- tions to each other, no matter wliat axes be assumed. In the second place, it might be urged that such ratios as those above given being accepted, there is no reason why we should at- tempt to express the relations of the prominent planes — those of humite, type II, for example, with simple numbers (see above, page 7). But, as has just been stated, the attempt to refer these planes themselves to other axes leads to disastrous results, while further, as has been shown, these planes are truly secondary and subordinate and bear no relations to other types of the species. This case has l)een dwelt ujjon at considerable length, because it was believed that theoretically the existence of such planes w^as a matter of some interest and importance, and because this single crys- tal offered opportunities for exact determination which did not exist to the same degree in any other case. Almost all of tlie twenty and more smaller crystals examined showed some of these secondary planes. In some cases, however, there was a tendency to rounded edges E. S. Dana — Chondroditt from the lllly-FoKfer Iron Mine. 81 without the foniiation of distinct planes, as is so common in many species ; and then nothing of course couhl he done. The foUowing tables, VII, VIII, TX, X, include the measured angles, with the symbols obtained, and the corresponding angles be- longing to them, for a considerable number of these minute planes. Those occurring on each individual crystal are arranged together, be- ing expressed by the same letter, and where the crystal has been figured, this is also indicated. Upwards of one-hundred of these planes were measured, and an attempt was made in every case to ob- tain a satisfactory index. It was concluded, however, to discard the Table VII. M C=z-i(010,. 1 Meas. Calc. i + 6^=24(201). Meas. Calc. — o 8-4 281 14° 6' 13° 41' 79° 2' 79° 16' — v, 11 11 T 5 5-11-4 25 14 25 34 73 29 73 7 -"2 8 16 3 '11 IT-16-6 33 50 34 5 64 40 64 28 -«3 13 5 4 4 52-65-20 37 7 37 19 60 0 59 49 -Si 3 15 Y 16 I6-15-10 46 43 46 56 57 38 57 22 -^ 5_ 5 357 46 4 46 21 81 35 80 49 -h 9_ 9 14 1 4-9-14 46 57 46 58 87 7 87 31 -«2 7_7 9 8 879 53 6 53 7 64 9 64 7 -*3 13 Y Y 236 '54 31 54 42 85 0 84 33 + h Y _7 TT Y 2-7-11 45 47 45 57 92 46 93 6 + t. 9 9 10 2 2-9-10 36 16 36 21 91 20 91 44 -n- 2 221 44 13 44 14i 56 45 56 42 -c 9 9 Y T 795 138 45 38 37 63 44 63 58 -^ 15 3 20-15-4 50 58 51 6 46 13 46 4 i' \-i Oil 32 34 32 27 79 55 79 54 +P^ 1- 157 42 0 42 17 109 36 110 7 -f > 4-20-17 ,29 1 41 29 44i 90 0 89 58 Trans. Conn. Acad., Vol. III. 1 1 .July, 1875. 82 E. S. Dana — ChondfrodiU from the Tilly-Foster Iron Mine. larger portion as untrustwortliy, retaining those which had given the best angles. It is not pretended that the symbols deduced are, even in the majority of cases, correct beyond question ; for the angles, while perfectly reliable in some cases, are in others somewhat uncer- tain, and for reasons already explained this throws still greater doubt over the indices which calculation may produce. Table VIII. 7 7 z' 6 5 11 11 2^ ¥ y 2^ 1515 4 14 Z^ 17_17 25 ^:- m 4 ^ 5 _5 c •= 1 7 _ 7 9 5 r^ 8_4 9 3 675 5-11-6 14-15 -4 4-17-3 2-10-11 641 259 579 689 C= w(OlO). Meas. Calc. 40° 0' 39° 52' 28 44 28 32 41 0 41 14 14 6 13 43 35 42 35 54 54 18 54 17 50 32 50 16 46 45 46 10 45 18 44 51 A= 0 (001). Meas. Calc. 65° 25'65° 51' 72 13 ,72 15 ■i 82 45 i82 40 83 55 83 44 55 44 55 28 84 37 42 45 42 56 55 48 55 29 59 59 30 84 40 The group of planes clustered about ^^ has already been de- scribed. It is interesting to note that in two instances analogous groups were observed, of which ^(l-i = Oil) was a member (see fig. 15), and in two other cases the common and prominent planes ^i^ (fig. 15) and vi^ (fig. 17) were found in the same relations. The angles obtained for these planes show conclusively the degree of dependence to be placed on those measured for the other planes. It will be remembered that in all cases these planes were exceedingly minute. The fact already mentioned, that all of these planes belong to the macrodiagonal series, may possibly be explained in part by the fact that it is uniformly that portion of the crystal (i. e., near 6', i l) which is exposed and well developed. There still remains the fact, wliich will be noticed by a glance at the spherical projection (plate xiii) and which does not allow of an analogous explanation, that E. S. Dana — Chondrodite from the Tilly-Foster Iron Mine. 83 Table IX. -■ e2=2t(201). Meas. Calc. ^ = 0 (001). Meas. Calc. £ i 1 4 012 047 51" 50' 47i-48 51° 50' 48 4 HH -3 1 ^ 2 025 58 18 57 51 j-« 8 _ 489 40 6 40 24 56° 42' 56° 59' — T 1 _ 5 4-510 55 45 55 47 46 7 44 18 -^ 11 11 10 Y 711-10 39 33 39 32 63 23 63 26 + 7r 1 5 Y Y 3-5-10 54 2 54 13 41 58 41 55 hH —nV 5 2~ 552 43 40 43 42 79 33 79 22 —n« 30 30-30-1 42 40 42 39 89 7 89 6 tliese minute planes almost always lie in the negative (lower) quad- rants ; this has necessitated the drawing of some of the crystals in an inverted position. The figures show, in addition to planes mentioned in the tables, some others for which no symbols Avere obtained. Table X. C = w(010). -e' =j.i (203). i 1-t 15 Oil 32° 20' 32° 27' 67° 15' 67° 16' — Wi 17 17 4-15-15 34 41 34 18 58 21 58 43 \-A 10 Y 6-17-10 26 33 26 25 63 29 63 37 1— 1 > 3 ^ -3 -W-, r' 134 41 34 42 3 52 45 52 18 >> 1 5 O — T Y Y 8 ^ 4-5-10 55 27 55 47 36 40 36 37 — 6 Y-* 15 , 289 37 10 36 57 56 27 57 21 -u, 13-^ 12 3-1513 29 22 30 10 63 45 63 2 -M2 u' 212-11 31 8 31 6 62 55 63 13 84 E. tS. Dana — Choiidrodite from the Tilly-Foster Iron Mine. Crystals of simjyler habit. — The crystals thus far described have all been of a more or less complicated character. But allusion has been made to some very simple crystals, which also deserve descrip- tion. The distinction is in most cases probably unimportant, though lielieved to be of interest at first. The simple crystals are uniformly large; they are so generally altered, and appear so differ- ently from their small brilliant relatives, that it was supposed that they differed from them at least in the purity of the original mate- rial, if not more essentially. One brilliant exception, however, to the general rule in regard to the altered condition of these crystals, in the form of an isolated crystal of faultless luster, and deep red color, about f of an inch in length, as well as numerous examples of transi- tion products between the altered and unaltered material, made it probable that all the crystals in question were originally of the same character. Some examples are given in figures 3, 4 and 5. The angles could be measured approximately only with a hand goniom- eter, but there is no question that they, as well as others, belong to type II. On one of them ±r was observed. In what has been said exception must be made in regard to the large coarse crystals, and crystalline masses, mentioned in the early part of this article, and which are made up of a more or less hetero- geneous mass of chondrodite, magnetite and sometimes rijjidolite ; some, at least, of these last, belong to type I. (See p. 25.) Twins. — The humite crystals of Vesuvius, as well as the Swedish chondrodite, has been shown by vom Rath to possess so great a ten- dency to twinning that it is a little remarkable that the contrary should be true of the mineral from Brewster. Figs. 20, 21, show the only method of twinning which has been found, as well as the only distinct twin-crystal. The axis of revolution here is the vertical axis of the crystal, and the composition-face the basal plane A. Unfor tunately the crystal in question was quite imperfect, and all that was available is shown in the figure. The plane H (/-?=iOO) gave no reflections, so that all measurements were made on e^(that is 201 and 201) ; in this case these planes were similar in luster as a result of the twinning. A revolution of the kind mentioned (in a perfectly symmetrical crystal) would, so far as this half of the crystal goes, have the effect only of making it holohedral, giving no re-entrant angles ; but, in case of any irregularity, it might give, as here, a re-entrant angle in the planes which are hemihedral in their occurrence. The measurement of the re-entrant angle here observed gave for *w2^to2, 10° 38' and 10° 40'; required 10° 39'. The other angles measured on the same crystal are given in the following table. E. /S. Dana — Ghondrodite from the Tilly-Foster Iron Mine. 85 Table XI. e2 = 2- Meas. I (201). Calc. e2=2- Meas. I (201). Calc. m- 641 37° 39' 37° 34i' 42° 54' 42° 57 ' m- 641 42 57 42 57 37 28 37 34+ m- 641 37 36 37 34i 42 55 42 57 m- 641 42 57 42 57 37 33 37 34i n' 221 45 45 45 45 56 42 56 42 v? 221 56 44 56 42 45 45 45 45 ri' 221 45 34 45 45 56 41 56 42 n' 22] 56 15 56 42 45 34 45 45 ml 223 44 45 44 38 70 16 70 20 iti 223 44 48 44 38 f'2 245 59 13 59 14 82 34 82 34i ri 241 64 3 64 2 0 210 33 52 33 50i 33 48 33 50^ <P 60-381 38 14 38 17 39 24 39 27| 641 641 (1 11 4 31 4 25 6 31 6 36| Besides the interest of the twin, a noteworthy fact in the crystal is the occurrence of the prism (/-2=210) the first time that any one of the vertical prisms has been ohserved in the 2d tyj^e of either humite or chondrodite. Tt lies in botli the zones e-(201), /^^ (2^1), and e2(22i), «- (22l) which answers sufficiently to determine what it is; and the result thus obtained is fully sustained by good measurements, as will be seen in the table. This plane is distinctly present on one side only of jB/ on the other side its presence is barely indicated. Its place here is taken by a well polished and conspicuous plane *, which is another striking instance of the peculiar nature of this species ; its position is indicated in the sphere of projection,* and the angles on e^ are given in the table. The index was calculated for each pair of measurements 38° 14', 39° 24 and 4'^ 31, 6° 31 (see in the table), and the results obtained were identical. From the first pair of measured angles 'i a.B was found to be 34° 30^' and from the second * a-S= 34° 31' (required 34° 32' 38"). The index obtained was 60-f^ or 60-38-1, and abnormal as it certainly is, it expresses the exact posi- * In the projection * is placed incorrectly in the negative quadrants ; it should be in the positive with + a, + n^, + r, etc. 86 E. S. Da7ia — Chondrodite front the Tilly- Foster Iron Mine. tion of the plane. It will be noticed that the four planes upon which the inclination of * was measured are so situated, that any variation from the true position in the index would sliow itself very conspicu- ously. The fundamental angles for * are as follows : */s^l(001)= 88° 51' 41'' ^^^(lOO)^: 34 32 38 */s (7(010)r= 55 28 49 In figure 16 a crystal, or portion of one, is exhibited which is holo- hedral. It is irregular in this respect, however, that r^ forms a re- entrant angle with r^. This is not a point of special importance, as an irregularity such as this is often observed ; but, in view of the crystal which has just been described, it is possible that here also there has been a semi-revolution parallel to the basal plane. A more interest- ing crystal, already once alluded to, is shown in fig. 4. It is con- spicuously hemimorphic, as far as the form goes. It is large, and admits only of approximate measurements, but there is no doubt but that the planes as given have been determined correctly. In view of the fact that a revolution parallel to 0 would produce just the eifect we have liere, and as such a twinning law has been shown to exist in another conspicuous case, it is altogether probable that this forms an ample explanation of what is observed. Another exactly parallel case is noted under the description of two crystals of the 3d type. The above described crystal was somewhat altei-ed, and so far imbedded in the matrix that any experiments as to its pyro-electrical character were out of the question. Chemical composition. — I am glad to be able to add here the re- sults of a chemical examination of the chondrodite of the 2d type from this locality, by Mr. G. \¥. Hawes of the Sheffield Scientific School. It obviously increases much the value of this memoir. An analysis by Mr. Breidenbaugh has already been quoted (p. 3). The material analyzed by Mr. Hawes consisted of fragments of crystals of the 2d type, selected with great care to avoid the pres- ence of any altered material. It had a deep garnet-red color and a brilliant vitreous luster. Its specific gravity as determined by Mr. Hawes was 3-22. Analysis I. Analysis II. Silica 34-10 34-05 Magnesia 53*17 53-72 Ferrous oxide 7-17 7-28 Alumina -48 "41 Fluoi-ine 4-14 3-88 99-06 99-34 E. S. JDana—Chondroditefrom the Tilly-Foster Iron 3fine. 87 Following- the view of Kammelsberg, that the higher values of each constituent are nearest to the truth, Mr. Hawes's analysis becomes as follows. For comparison the results obtained by vom Rath for 2cl type crystals from Vesuvius and from Sweden are added. Chondrodite. Brewster, N. Y., Hawes. Swed Silica 34-10 . Magnesia 53-72 Ferrous oxide 7 '2 8 Alumina 0-48 Fhiorine 4-14 99-72 Silicon 15-91 Magnesium 32-23 Iron 5-66 Aluminum 0-26 Fluorine 4-14 Oxygen 39*78 HUMITE. V. Rath. Vesuvius , V. Rath. 33-96 34-02 53-51 59-23 6-83 1-78 0-72 0-99 4-24 2-74 99-26 98-76 15-85 15-88 32-11 35-54 5-31 1-38 0-38 0-53 4-24 2-74 39-58 41-54 97-98 97-47 97-61 Transforming the iron into an equivalent of magnesium, as also the alumina (2Al=:3Mg), Mr. Hawes obtains further: Silicon 15-91, Magnesium 35-00, Fluorine 4-14, Oxygen 39-78, From these values a formula is deduced, which is essentially that of the Swedish mineral according to v. Rath, 20(Mg Si20g)+ MggSigFl^g. The close correspondence between the three analyses in the above table is certainly very remarkable. It would have been extremely interesting to have added an analysis also of crystals of the 1st and 3d types; but, as will be appai-ent from what follows, the material was not to be obtained. In completing the description of this variety of the mineral, in general it may be repeated that it occurs usually in narrow veins, and when free from alteration has iiniformly a deep garnet-red color. A cleavage such as exists in humite (parallel to the basal plane) and has been observed by Kokscharow on chondrodite from Pargas, could in no case be discovered. The fracture is always conchoidal, and the only thing that suggested cleavage was the laminated structure of the massive specimens described, and the fractures parallel to e^ which were quite conspicuous on two or three specimens. The ma- terial in hand Avas not well adapted for optical determinations- 88 E. tS. Dana — Chondrodite from the Tilly-Foster Iro7% Mine. but some interest in <>• results liave been obtained whicli are described in the closing pages of this paper. 2. Description of (Jri/stals of Type III. Since it was shown by Kokscharow that the Pargas chondrodite was identical with the second tyjie of humite, it was natural to ex- pect that further investigation would prove the existence also of the two remaining types. Up to the present time that expectation has been unfulfilled, and it has been reserved for the Brewster locality to give this confirmation of Scacchi's interesting discovery. The crystals of the 3d type are exceedingly rare, three or four specimens being all that have thus far been foiind, and from these only two individual crystals could be obtained which allowed of measurement. Fortu- nately these two crystals are very satisfactory, bemg small and bril- liant, and establish the fact as well as a hundred could do. Figures 11 and 12 show one of the crystals, and figure 13 the other. The appear- ance of the first crystal is best shown in the second of these figures. As will be seen, the planes are the same as in humite, and they are for the most part hemihedral and situated in the same way ; i.e., they in- clude+p',+p', +//, and -p\-~p\—fj\ ^\\(\.—v\-y\-v\—v' as also 1^,1^,1*. Ill the n{v) series the planes are distinct in the negative half, but the crystal is incomjjlete, and it is a little uncertain whether the -|- series should not in part be added in the symmetrical drawing fig. 11; on humite. III, these planes are both positive and negative. No brachydomes are visible, the edge being rounded and rough. ni~ of Scacchi may also be present, but that is a little doubtful. The second crystal is of very different form, and Avhile the first was affixed to the rock so that only one-half was developed, this one was imbedded in bnicite, and entirely free in it. It was perfectly formed on all sides, being almost as perfect as the projected drawing, with the exception, however, of the acute (brachydiagonal) edge, which was mostly broken. When only the upper part of the crystal is considered, it will be seen that the hemihedrism is like that in the other case, ex- cept that (f is holohedral. For macrodomes there are /'(f-7^023), z-(l-73=011), /'(2-7— 021), /'(4-7=:041); the last has not been observed on humite. On measuring the planes below it was found that they were not distributed as was expected in accordance with the mono- clinic character of the crystal ; instead, either exti-emity of the brachy- diagonal axis was diffei'ently developed. This is clear in the figure, it being but a more complicated repetition of what was observed in one jE ^S, Dana — Chondrodite from the Tilly-Foster Iron Jline. 89 of the very simple crystals of the 2d type (see fig. 4 and page 18). There are present also at one extremity it ^'(t"^^^'^^''^)? though the plane could only be approximately measured. This is probably also to be explained as having resulted from a revolution parallel to the basal plane. The crystal was very small and not at all adapted to expei*i- mcnts having in view the discovery of any proper hemimorphic developiuent. The angles measured on both these crystals are con- tained in the following tables. Table XII. Chondrodite. Hiunite. Calc. ^=0(001). Meas (XX). Meas (XXI) V. Eath. Calculated. £ 4 407 143° 20' 9" 143° 11' 29" C^ 4-1 041 100 1 7 l^ 2-1 021 109 27 35 109 27 54 I- l-l Oil 125 14 49 125 15 18 il 2 — -i 3 023 136 40 4 136 40 34 P' A-2 11 2811 131 25 57 131' 46' 131° 24' 131 24 49 P" 1-^ 289 125 50 6 j 125 37 I 125 47 125 48 125 49 0 p= 1-^- 287 119 19 18 j 119 35 ] 119 15 118 36 119 18 19 P' 1-2 5 285 111 51 38 j 111 44 (111 49 112 0 111 50 50 P' 8 _ 283 103 32 4 103 41 103 38 103 31 33 p« 8-2 281 94 35 15 j 94 31 I 94 13 94 48 94 35 4 v' 4 T 447 132 17 48 132 16 132 16 43 V^ 4 ~5 445 123 1 8 122 32 123 0 8 ^3 4 443 111 18 7 111 5 111 17 23 V^ 4 441 97 24 20 97 29 97 24 3 Unfortunately the inclination to C on no one of the pyramidal planes could be measured with perfect accuracy ; the measurements are good, yet not entirely trustworthy. These planes, though brilliant, are uni- formly fi-actured in the manner already explained, and this made all Tbans. Conn. Acad., Vol. III. 12 July, 1875. 90 E. S. Dana — Chondrodite from the Tilly-Foster Iron Mine. Taijle XIII. Chondrodite. Calc. C=i-l (010). Meas (XX). Meas (XXI) i* 4-1 041 169° 58' 53" 170° 2' (? 2-1 021 160 32 25 160 36 i^ 1-1 Oil 144 45 11 144 47 «! 2 023 133 19 56 133 (ap.) p« n-^ 2811 132 56 12 j 133 13 I 132 38 132° 46' P' 1-9 9 289 137 26 13 j 137 36 1 137 27 137 21 P" f^ 287 142 22 50 j ]42 35 1 142 20 142 38 P' 1-^ 285 147 28 24 147 26 P' 1.-2 3 283 152 1 44 152 2 P' 8-2 281 154 53 20 155 0 154 58 v' 4 y 447 123 31 12 v' 4 445 128 5 52 V^ 4 "3 443 133 16 51 133 57 V* 4 441 136 51 ' 136 42 these angles a little uncertain. The macrodomes in one case gave good measurements; and making use of the best of them, C/\i^{l-i) r=144° 47', and also the same prism as in type II, after the analogy of humite, the angles were calculated throughout. It was found, however, that wherever trustworthy they corresponded so closely to what was required to satisfy the ratio of 10:9 for the vertical axes (asserted as approximately true by Scacchi, i. e., 5 : 9 for his axes, and finally proved rigidly by vom Ratli), that the calculations were made on this supposition. The calculated angles, as now given, conse- quently have as their basis the prismatic angle Iy,,I=z94° 44' 14" and the macrodome angle C^i'=l^-i° 45' 11 ". The coiTcsponding parameters are : a (vert.) = 1-41512; b = I ; c = TOSBSO. E. JS. Dana — Chondrodite from the Tilly-Foster Iron Mine. 91 Very little further can be said in regard to the crystals of tlie 3d type. Those observed had a somewhat diiferent color from those of type II ; that is, the color was more yellowish, less of a pure garnet- led — though this may be accidental. No analysis was possible of coui'se; and even the sjjecitic gravity was out of the question also, for the one loose crystal, in addition to its small size, had imbedded in it a still smaller crystal of ripidolitc, making any gravity determina- tions obviously uurelial)Ie. The method of occurrence was much like that of the brilliant crys- tals of the second type ; that is, they w^ere found implanted on the massive rock adjoining small veins. The associated minerals Avere magnetite, ripidolite in clear transparent crystals, and, probably as a later formation, brucite. 3. Description of Crystals of Type I. The occurrence of large, coarse crystals of quite impure chon- drodite, imbedded in the massive material, has already been de- scribed ; these belong, at least in part, to the first of Scacchi's types. As has been remarked, the crystals of this character do not often admit of exact determination, but in two cases they were so good as to allow of their crystallographic relations being accurately made out. The accompanying wood-cuts, figui'es 22 and 23, give faithful representations of their appearance and size. RU «^ J^l R2\ / R3 /r R^ \ *L__b4 J R^ \ — -— ^-,^_^ 1 r. ^^=Sh 1 ,4 ^^z^ R^ R^" Fig. 22. Fig. 23. It will be seen that they are both quite imperfect, and it was on this account that no attempt was made to make a symmetrical drawing of either of them. In each case the crystals w^ere so distorted as to give a sharp edge between the diagonally situated pyramidal planes; this furnished an opportunity for relial)le measurements. The plant's 92 JiJ. 8. Dana — Chondrodite from the Tilly-Foster Iron Mine. were, of course, destitute of all luster, but they were mostly smooth and large enough to allow of the convenient use of the hand gonio- meter. R"^ on 7^" (behind) gave measurements varying, in a series of trials, from 78° to 79", required 79° 4'. 7t" on R' (behind) gave 62^, required 63° l'. i?" on R^ (l)ehind) gave 72 , required 71 174- Zil' on R" (behind) gave 72 , required 71 17^. These angles on both crystals were identical within the allowed error of observation (say 30'). The above are the best angles afforded by any of the planes. It is entirely impossible to refer these angles to any of the forms of the second type. When compared with the third type, it is seen that on making the supposition that R' and W (front) are p^ and p* respectively, and R"^ and R^ (behind) are p* and p^, we obtain for: p\\ p' (behind)=77° 12'; pV p' (behind)~65'^ 6'; p'^ p' (behind)=:70° 32'; pV p' (behind)=:7l° 37'. It will be seen, by comparing these with the ])reviously given angles, that the measured angles correspond much better with the iirst type, and my confidence in them is so great that this would alone be re- garded as sufficient to establish the point ; and that, without refer- ence to the fact that tlie supposed method of occurrence of the third tyj^e planes is contrary to all the laws of the species. The decisive proof is derived froTU the fact that both crystals are certainly holohedral, the planes on both sides being similar with the exception of R\ and there is nothing of the obliquity which is ob- served in the hemihedral forms. The measured angles of C on _Z?\ right and left, were identical, though not obtainable with exactness ; the measurements gave 152^° -154° : this is also true for C on R\ right and left,= 140i°-142^°. 3 In the first crystal e/'::^ i (035) occurs, and in the other J'^ o (l-7=r01 1). The occurrence of C is also to be noticed, as it is rare on humite; in fig. 23, the oscillatory combination of 7^"* and R" will be also observed. The following table includes the most important angles for the oc- curring })lanes, calculated from the fundamental form of the second type on the assumption that the lateral axes are equal, and the vertical axes have the ratio of 14 : 15. The measured angles are also added thouo"h only approximate ; in the form given they were obtained imme- diately from the measurements over the top of the crystals (see above). E. S. Dana— (Jhondroditc from the Tilly-Foster Iron Mine. 93 Tahmc XIV. Chondrodite. Humite. C=M(010). Calculated. .1= 0 Mens. (001). Calc. V. Ratli. Calculated. J« l-l Oil 145° 43' 44" 124° 16' 16" 124" 16' 45" J> 3 035 138 38 38 R' f^ 3-6-10 129 12 5*7 135° 135 53 35 135 52 23 B? Q 4-^ 368 134 28 38 129i 129 32 3 129 30 52 W 1-2 122 140 30 10 121i 121 45 28 121 44 23 R* 1-^ 364 147 6 34 112 25 28 112 24 37 R5 3-2 362 152 49 49 101 101 39 30 101 39 2 The two crystals described are the only ones which could be posi- tively identified. It is very probable, however, that of those found others also belong here, as they have much the same appearance and habit. These crystals are all considerably altered, being generally soft enough to be cut with a knife, and for this reason a chemical analysis would be of little value. The color of the crystals is gray to grayish-yelloM', and the material of which they are composed is never pure, and often quite heterogeneous. In this respect they recall the bi'own crystals described by v. Rath as occurring at Nya-Koppar- berg. Whether brilliant crystals of the first tyjje exist, as they do of the other types, must be left for the present undecided. 4. On the Optical Properties of Chondrodite. In the preceding pages the question of the orthorhombic or clino- rhombic crystallization of the chondrodite has not been discussed. In fact, nothing was detected by the measurements sustaining any other conclusion than that of Scacchi and vom Rath, that the crystals were fundamentally orthometric. Still the hemihedral character of the second and third types seem to point to a clinometric form, and this is apparently supported by the optical characters obtained. The ma- terial available for optical investigations was very scanty, and, with the exception of one crystal, poorly adapted for the purpose. The crystal referred to was, properly, but the fragment of what was originally a specimen of considerable size and beauty ; 94 E. S. Dana— Chondrodite from the TUhj-Foster Iron Mine. when unbroken it must have been nearly an inch in length. In the condition in which it was found it showed only the brachydomes e' ande^, with the pyramids ?^^, n^, and m^ ; it had the deep garnet-red color of crystals of the second type, and with the exception of the uni- versally i^resent fractures was perfectly clear and transparent. On the optical jDroperties of the mineral hi question, we have, as far as I have been able to find, no information except what has been giveu by DesCloizeaux, Manuel de Mineralogie, 1862, p. 141. He says : " Double refraction energetic ; positive bisectrix normal to ^ * ; optic-axes situated in the jdane parallel to the base ; divergence in oil for red and yellow rays, 82° 14'. Dispersion almost nothing; Guided by the above, a section was cut from the crystal described, which was pai-allel to C', i. e., perpendicular to the brachydomes pres- ent. The examination of this section showed : i st, that the acute bisectrix is normal to G (i-i, 010) ; 2d, that this bisectrix is positive ; 3d, that the optic-axial angle is large, the axes being seen only when oil is used ; but 4th, that the axes do not lie in the hasal jjlane, but in a plane making an angle of about 15 t° with it. This last point was so unexpected and anomalous that every effort was made to explain the measiirements in soine other way, but with no success. The planes on the crystal had been carefully measured, before the slicing, and the angles agi-eed perfectly with those of type II for the planes mentioned, so that it was impossible to assume that the crystal had not been correctly put into position. By means of a staiiroscope, made by Fuess in Berlin, after the excellent pattern of Groth,* the posi- tion of the two axes of polarization, as referred to e^, and also to e^ in plane C, were carefully determined. The measurements were repeated twenty times, the error arising from an imperfect adjustment of the Nicols being eliminated in the usual mannei*. The result was as follows : Supplement angle made by the plane of the axes — with e^(|-^=r203), 18° 9' ; hence with the basal plane, (J, 25" 50'. withe^(2-^=201), 45° 9'; " " « " 25° 46'. In order to confirm these results, other crystals were sought, which would admit of like determinations. None could be found which would serve for measm-ing the axial angle ; but two small ones, on which the plane C was naturally developed, proved to be clear enough to allow of measurements with the stauroscope. The first *See Pogg. Ann., cxliv, 34, 1870. K S. Damt — Chan drodite from, the TilU/-Fostei' Iron Mine. 95 nloiie gave accurate results; on it tlie angle of the same plane with 6"(|-?=i206) was deterniined with equal care. The results were : 4° 55' for the angle with ea; and hence 25° 59' with G. The agreement with the angles given above is as close as could be desired. In the other case, the rather rare plane B (^-^r=100) was pres- ent ; the crystal was minute, however, and the determination only approximate. It was found that the normal to the axial plane made with B an angle of 65°-V0°, and hence with the normal to the basal plane 20°-25°. With so ample confirmation the point made cannot be even ques- tioned, and it remains to reconcile it with the crystallographic proper- ties of the species. It will be seen at once that the position of the optic axes is totally at variance with the accepted orthorhombic character of the crystals ; but it conforms to the rule for monoclinic crystals, as one axis of polarization is normal to the plane of symmetry C, and the others lie in it, or in other words, the optic-axes lie in a plane per- pendicular to the axis of symmetry. The angles measured and cal- culated, given in the various tables, show that the variation from the rectangular type, if it really exist, must be very slight, as the agree- ment between the angles measured and those calculated on the assumed prismatic basis is very close — it being remarked that some con- siderable variations in the angles given in the tables are amply ex- plained by the imperfection of the crystals. Note the angles measured for m^/\m^ on the twin crystal described on page 18. It was not to be expected that the variation in the optical character of the crystals would be so decided in view of the slight divergence which is possi- ble in the crystalline form. I reserve for the future the careful re- vision of the angles of this species, when I shall hope to be able to command a more abundant supply of satisfactory material. It may be added that the hemihedral character of the second and third types of humite long ago suggested the idea that they were oblique inform ; but all the crystallographic investigations thus far have seemed to deny this. In the Mineralogy of Brook and Miller, the form is made oblique, but this seems to be due to a misunderstanding of the planes occurring on the crystals. It would have been interesting to extend these observations to the two remaining types, but the material did not allow of it. It was also desired to investigate the same subject for humite, but, though some good specimens are to be found in the Yale cabinet, there were no satisfactory crystals to be had, and the matter is left for others, who have a larger choice of specimens. The axes as already men- tioned do not appear distinctly except in oil ; in the first-mentioned 96 E. S. Dana — Ghondrodite from the Tilly-Foster Iron Mine. section they admitted of good measurements. The mean of thirty determinations of the angle for red rays gave — 2Ht>'=88° 48' : the extremes being 88** 36' and 89° O'. With a yeHow light (sodium) the angle was essentially the same, but tlie mean was 10' or 15' smaller, which would indicate that the dispersion is /3]>?^, but the matter cannot be considered to be beyond doubt. No other dispersion was observed, that is, none parallel or perjDcndicular to the plane of polarization. The index of refraction of the oil employed, as determined by Pro- fessor Wright and myself, was 1"466. In conclusion, I have to expi-ess my very great obligations to Prof. Allen for his kindness in giving me free use of all the specimens in his valuable cabinet. Both of the crystals of the third type, as well as several others mentioned, came from his collection ; in fact it was Prof. Allen who first made known the special interest connected with the 'locality. To Mr. Cosgriff, the superintendent of the Tilly- Foster Iron Mine, I am also much indebted for his uniform kind- ness and courtesy to me at the several occasions when I have visited the mine ; as also for the gift of several fine specimens. IV. — Ox THE Til. vxscEN DENTAL CiKVEs s'mi/ smmy=asu\XHiunx-\-b. With Plates XIV— XXXVII. By II. A. Newton and A. W. Phillips. 1. Algebraic curves have been studied hitherto more than trans- ceudeutal. A few of tlie latter have beeu giveu in the text books, but attempts to classify the numerous varieties of transcendental curves have been rare. From the form of a transcendental curve it is not easy to state an equation that can represent it. The simpler inverse problem of describing the curve from the equation is naturally the first to be undertaken. The forms that result may, when compared, suggest the solution of the direct problem. We have thought it worth while, therefore, to select for study a single one of the numberless transcen- dental equations, and to exhibit a few of the very many plane curves which that one equation furnishes. 1 he equation selected is, sin y sin my =l asinx sin nx-{-b, ( 1 ) in which there are four arbitrary constants a, b, m, and n, with two coordinates, x and y. 2. We assume that ui and u are each less than unity. If either, for example >;/, is greater than unity, we may change the unit for y in the ratio of 1 :m\)j writing y'=:zmy. The first member of Eq. (l) . 1 , . . , , . 1 then becomes siny' sin — y', where the coefficient of y is — , which is less than unity. In the equation thus changed, we have assumed in our figures the units for x and y equal, and the axes rectangular. The effect of a different supposition in either particular can be readily understood. 3. Curves xchose equations <ire y=zasi)ixsinmx. It was found convenient to draw several auxiliary curves whose equations are of the form, y=zaii\\\x^mnix. (2) A convenient arlutrary value being assumed for a, to m was given in turn all the values of the proper fractions, which, reduced to their lowest terms, have denominators less than 12. The forms of these curves are shown on plates XIV and XV, excepting a few in which m has 11 for denominator. In fig. 37 is shown the beginning of the curve when m has the irrational value s/\. The axis of x is drawn Trans. Conn. Acad., Vol. III. 13 August, 1875. 98 Neioton, and Phillips on certain Transcendental Curves. in the figures. The origin is the point at the left of each figure where the curve touches the axis of x. P p' It will be convenient at times to put m= — , and nz=z —, where/?, ^, p', and r/, are integers, and the fractions are reduced to their lowest terms. 4. Properties of the curves of Eq. (2). By inspection of the curves on plates XIV and XV, and of their equations, we readily deduce the following properties : a. The value of y is not greater than a. h. When either x or inx is a multiple of ;r, ;y=:0. c. There are maxima or minima values of // when wi tan ,r= — tan ma*. (/. When ni is rational the values of y repeat ; after qn if p and q are both odd ; after 'Iqn if either^:) or q is even. e. When m is irrational the curve does not repeat its form. /'. The curve is symmetrical about the axis of y, and about an axis through the middle point of each cycle. g. If p or q is even., the curve is symmetrical about the point y=0, qn ^' h. There are, in each distance -Iq-n: along ,<■, p-\-q maxima, and an equal number of minima values of y. 5. The value of y in Eq. (2) may be regarded as made up of two a a parts, since y=La&mx %\nnix=z~ go^[\ ~m)x — — cos(l-|-m)a.'. In fig. 22, where mz=L^, these parts are sej^arately shown. The continu- a ous line represents the curve yz=: — qo^[\—^)j\ having one com- ])lete oscillation in a distance of IQn along ii-. By laying ofl" below and above this curve the second part of y, that is— — cos(l -|-f)^, we have the curve y=.a%mx sinf.«. 6. Use of the auxiliary curves, Eq. (2). To draw the curves from equation (1), even after all the usual devices for saA'ing labor have been employed, requires the frequent solution of equations of the form sin a; sin?;? 35:= c. This equation gives a set of values of x for each cycle of the curve. To find each value of x requires a solution l)y trial and error, a very simple process, but when often repeated quite tedious. By the curves figs. 1-37 carefully traced on cross- section paper we may by merely running the eye along the line y=.c NeuitoH and PhUUpn on certain IVanscendental Curves. 00 obtain by iiispoctit)ii all the values of x to a sutHcieiit degree oi" accuracy. 7. Equation (1) when a=zb=zO. The ecjuatiou siny sin wymO, ['A) consists of the two equations sinyzrO, and sin ?>iy=:0, and is satisfied by the values i/:=l7r, and i/ii/=f7r, where / is 0, or any integer. In fig. 60 the horizontal lines belong to the equation sin // sin |- ;y^=0. They consist of two series, one at intervals of ;r, the other at intervals of 2^7r. Tf through the intersections of the curve in fig. 25 with the axis of .(• there be drawn lines peri)endicular to that axis, the lines for smx sin |a*r=:0 would be obtained. The heavy lines of fig. 60 I'epre- sent double lines, corresponding to points of tangency in fig. 25. 8. Equation (1) ivhere a=0. The equation (1) becomes by mak- ing a=0, and for convenience changing the axes, sina'sinwia-^J. (4) This does not contain y, and therefore represents straight lines parallel to the axis of y. If the straight line y=b be drawn parallel to the axis of a^ to cut the curve y=s\nx sminx, and through the several points of intersection straight lines be drawn parallel to the axis of y, these lines will evidently be those represented by the equa- tion since iimutxz=.b. In fig. 60 the vertical lines rei»resent the equation sinx sm^x:='f. If the curve in fig. 26 be cut by a line parallel to the axis of x and distant from it two-fifths of the largest ordinate, the intersections will correspond with the intersections of any horizontal line in fig. 66 by the several vertical lines.* 9. Equation (1) iohere a=\, b=0, m=zn—\. The equation sin y sin ?/=sin x sin x (5) becomes sin y= ±sin .'■, or ;/z=Itt ±:-'', I being 0, or an integer. The cvirve consists of two series of parallel equidistant straight lines, the one parallel to //=.>', the other to y=-x, and both cutting the axes at intervals of rr. The locus is represented in fig. 38, where the origin is any point of intersection. 1 10. Equation (1) v^here a=\, b=0, m.=n= — . The equation sin y sin— V— sina; sin —x (6) q' q is one of the simpler examples of equation (1). * The unit of abscissas in plates XIV and XV is smaller than in the other plates. 100 Nev)ton (Did Phillips on certain Tnuiscendental Curves. a. It is satisfied if y^=-'X.^ or if y-=. — x. Hence the two straight lines y= zb'*' form part of the locus of equation (6). h. If 'llqn-^x l)e put for cc, / being an integer, the equation is un changed, whether q be odd or even. c. If q be odd the equation will be unchanged if lq7r-\-.i' be jnit for x. d. The curve repeats itself to the right and left, and also above and below, at intervals of qn \i q is odd., and at intervals of Iqn if q is even. e. Straight lines parallel to yr=±./', and cu.tting the axes at intervals of qrr, or 2q7t, according as q is odd or even, belong to the locus of equation (6). f. These straight lines divide the infinite plane of coordinates into equal squares for a given value of q. Each square contains a similar and equal portion of the locus. If q is odd, that portion is not always similarly placed, ibr it may have two positions with respect to an axis. g. If q is even, isolated points at the centers of the squares {f) belong to the locus. h. The equation (6) is satisfied if sin ^=:0, and 9my=.0. Hence the locus of (6) passes through the angular points of all the squares formed by the lines sin a;z=0, and sin y^O (Art. V.) i. A few curves representing equation (6) are shown in figs. 40-4'?. The axes are not drawn. Any point of intersection of straight lines that is sui'rounded by an oval may he taken for the origin. The several propositions of this article will be more easily un- derstood by inspection of the curves. 11, Equation (1) lolien a=l, bz=0, ni=:n:= /- . In this case the equation becomes, smy8iu±--y=:iiinx't^m^.x. (7) q' q The properties of the curves of equation (7) are in many respects like those of equation (tj). a. The two straight lines y:=: ±cc belong to the locus. b. If p and q are both odd, the equation is unchanged, if y or x is increased or diminished by multiples of qyt. c. If either /» or q is eve)t, the equation is unchanged if y or x is increased or diminished by multiples of 2q7r. d. The curve repeats in the direction of either axis ; at intervals of qTT if p and q are both odd, at intervals of '2q7r if either jd or q is even. Neioton and Phillips on certain 'Iranscendental Curves. 101 e. Straight linos itanvllel to y=i^.r and cutting tlu' axi's at intervals oiqn, or 2fy7r, according as ]>-\-q is even or odd, belong to the locus of equation (7). f. Tliese sti-aight lines divide the plane of coordinates into ecjual squares for any given value of m. Each square contains a similar and equal portion of tlie locus, though not always siniilarly placed, g. Eqixation (7) is satisfied if sin.v sin?y^T=iO, and siny siii///y— 0. Hence the locus passes through all the angular points of the rectangles formed by these two series of parallel straight lines (Art. 7). h. If 2^-\-Q is ^<^<^ isolated points appear, belonging to the locus, at the centers of the squares. i. The maxima and minima values of y are determined by the equa- tion £- tan a*:^ — tan^^a? (Art. 4, c). This equation represents q q straight lines parallel to the axis of y. There are 2(jij) + (/) of the lines (Art. 4, h) in an interval of 2q7T. j. The same equation in y gives the maxima and minima values of x. k. These equations are also the conditions of the isolated and double points. Hence there can be isolated or double points only at the intersections of the lines i- tan xz= — tan ^-x with the q q lines -L- tany=: — tan i-y. q q I. The propositions {i), {J), and (k) hold equally true for any values of a and b in equation (1), and there are similar properties if m is not equal to n. 771. The figs. 48-65, 68, and 70, represent curves belonging to equa- tion (7). Any point where two straight lines meet, and that is surrounded by an oval may be the origin. 91. Tf through the double points on the line y=:x vertical and horizon- tal lines be di'awn, these lines will pass through all the points of maxima and minima ordinates and abscissas. By their intersections they will mark all the possible positions of double points for any values of a and b. 12. Equatio7i (1) lohen a=l, b=iO, m-=.u=:an irrational number. The equation sin y sin \/i y=sin x sin \/i ;r, (8) represents a class of curves tliat do not repeat their forms but change 102 N^ewtoii CDid Phillips on certain Transcendental Curves. continually throughout the plane of coordinates. The curve is sym- metrical about either axis, and also about either line yz=. i.e. These two lines belong to the curve. The origin and a portion of tlie curve, principally in the first quadrant, are given in fig. 67, ])late XXII. 13. Equation (1) when «— : — 1, JirzO, ^/i=in-= — If (j is eiwn the equation sin y sin — y =: — sm x sin —x (9) q q merely changes the sign of the second member if we substitute 5';r -fa; for X. Hence the curves in figures 40, 42, 44, and 47 represent equa- tion (9), when q is eium, the origin being at an isolated point. But if q is odd we obtain new forms which have these properties. a. The origin is an isolated point. b. If q=l, the locus consists solely of points (fig. 39). c. li q=z3, each point is surrounded by one closed curve (fig. 93). d. If q=5, each ^aoint is surrounded by two closed curves (fig. 74). e. The resemblance of these figvires to parts of figs. 40, 42, and 44, and the law ol' their formation makes it unnecessary to give further examples. f. A dot and four suiTounding closed curves in fig. 47, would fairly represent the element for equation (9), when ^1=9. 14. Equation (1) when az=. — 1, 5=0, m:=.n=. ^L. Curves whose equations are of the form sin wsini^y=: — sin x sin ^x (10) q q are shown in figures 69, 71, 99, and 108. There are no straight lines belonging to the locus. The origin is at any one of the isolated points. The first two are placed beside figures 68 and 70 for ease in comparing. The following propositions of Art. 11, for equation (7) apply also to equation (10), without change of terms, viz : i, c, c7, g.^ /,J, and h. 15. Equation (1) when az=.\^ l>=0, m=z], and ji= i_ . The figures 9' 76-79, and 81, represent curves whose equations are siny sin2/=sin£«sin-£--a;. (11) 9' Newton ant/ JViiUips on. certain Transcendental Curuefi. !();< Ill tlu' tlirection of// tliev repeat at intervals of tt. In (ln' direction of .(• they repeat at intervals of qTT^ or 2(/'7r, according as p'-\-q' is even or odd. Fig. 80 gives a similar cnrve except that r/= — I. 16. Equation (1) idieji a=^\, b^=.0^ in-z l^^-Mu\n,=:i-L. The eqna- q q tion (11) is a special case of the equation sin y sin ±- y := sin a; sin :^ a-. (12) q q' Examples of curves from equation (12) are given in figures 82-91, 123, and 141. The number of different curves that this equation gives us is quite large, even if q and q are limited to small numbers. If 11 is the maximum value of q and q' , the number of inde]>endent curves belonging to the equation is nearly a thousand. Equations (5), (6), (7) and (11) are special cases of (12). IV. Further consideration of the curves of equation (12). a. If the parallel straight lines sin,r sin:^ir=0 be drawn (Art. 7) the plane of coordinates is divided by those lines into portions. When two lines coincide the portion between them may be regarded as real but infinitessimal. In crossing any of these lines the sign of the second member of (12) changes from plus to minus, or vice-versa. h. In like manner in crossing any of the parallel lines sin y sin ^y=zO, ^ q the sign ot the first member changes. c. The lines dny sin ^(/--O, and sinic sin^,r=0, divide the plane q q into rectangles (some of Avhieh are infinitessimal). The curve of equation (12) passes through each of the angular points of these rectangles. d. Since the signs of the two members of (12) must be alike the curve passes at any angle of a rectangle into the rectangle vertically opposite. It passes from a rectangle only at the angles. e. If, however, any rectangle is of infinitessimal breadth and finite length, the curve at its extremity becomes tangent to the line that limits the infinitessimal parallelogram. /. If a rectangle becomes infinitessimal in both directions, the curve has at that point an isolated or a double point. f/. The horizontal and vertical lines of fig. 148, and the rectangles formed by them, illustrate the above propositions. The con- 104 Nev:iton and Phillips on certain Transcendental Curves. tiiiuous closed line represents the curve of equation (12) in this case. The axis of y is the heavy vertical line, and the axis of X the upper heavy horizontalline. These heavy lines are double lines. h. Several of the propositions of Art. 11 apply to equation (12) with p . p' evident modifications. If — is not equal to — „ there are no straight lines belonging to the locus. i. We may regard the plane of the curve as divided into equal rectangles by lines parallel to the axes, the altitudes of the rectangles being qrr, or 25-7?, according as p-\-q is even or odd, and whose bases are q'n, or 25-' tt, according as p' -\-q' is even or odd. The curve (12) repeats itself in each of these rectangles without any variation, through the whole extent of the plane. j. The origin of (12) is a real double point. 18. Effect of a change of the value of a in equation {\),ichenb=iO. The effect of a change in the value of the coefficient of the second member may be observed by comparing some of the figures : for example, figs. 38 with 39 ; figs. 41 with 93 ; figs. 45 with 72 and 73 ; figs. 77 with 80; figs. 43 with 74 and 75 ; figs. 123 with 131-135; figs. 136 with 141 and 145. 1 9. The effect of the change of this factor can be better observet). in the simpler equation sin y= ^ sin x, (13) where k represents a as assuming several values. Figure 130 repre- sents a faisceau of curves for equation (13). The origin is the nodal point near the lower left hand corner of the figure. Let k change from — cc to -j- 00 . a. If ^'rroo , we have the vertical eqiiidistant straight lines. b. If k=. — 2, we have the curved lines represented by uniform fine dots. At the origin it is tangent to y= — 2x. c. If A:= — 1, we have the straight lines in which dots and strokes alternate. d. If /<;— — ^, we have the continuous curved lines. e. If A'-nO, we have horizontal straight lines. f If k=:^, we have the heavy dotted curved lines. g. If k=zl, we have straight lines of which y=x is one, and the others are similarly marked. A. If k=z2, we have the curved lines consisting of a stroke and three dots alternating. Newton and Phillips on certain TransceMilental Curves. Kif) i. If X-=-|-a:,we liaA'e vertical lines iigaiii. The curve is at the origin always tangent to yz=ikx. The faisceau has nodal points wherever .r and y are both multiples of n. 20. If we consider in like manner the faisceau of curves sin y sin f y=:A;sin x sin fit', ( 1 4) for various values of k (fig. 148), we shall find similar but more com- plicated changes. The origin is the intersection of the heavy lines near the top of the figure. The figure represents the loci for six values of Jc, viz: oc, —1, — |, 0, +1, and -\-2. Each of the six loci passes through each nodal point, if isolated points be counted as branches of a locus. a. For kz=. oc, we have the vertical straight lines. The heavy line is a double line. b. For k=. — 1, we have the uniformly dotted curves. c. For k-=i — ^, we have the curves represented by strokes and four dots alternating. d. For X-=:0, we have horizontal straight lines, the heavy lines being double. e. For A,=rl, we have the continuous curves (compare fig. 14V). f. For A'=2, we have the curved lines consisting of a long stroke and a short stroke alternating. By removal upward or downward a distance of Stt, the curve (b) coincides with (e). In general any one of the curves by such change coincides with that one for which k has an equal value with opposite sign. 21. We may in like manner obtain a faisceau of curves from the equation sin y sin my-^k sin x sin nx-\-b, (15) by giving to k different values. The curve will be the horizontal lines siny sin myz=b (Art. 8), if kz=zO. It will be the vertical lines sin x sin mx=0, if kz=0. For other values of k, the curve will pass through all the points of inter- sections of these series of straight lines. Figure 66 represents (with the axes interchanged) the vertical and horizontal lines in a special case. The lines of maxima and minima values of x and y, and the pos- sible positions of double points (Art. 11, i, ./, k,), are independent of k and b. The origin is not upon the curve if k and b are finite. Trans. Conn. Acad., Vol. III. 14 October, 1875. 106 Newton and Phillips on certain Transcendental Curves. 22. Chayige of value of h in equation {!). It remains to consider the effect of a change in the constant b in equation (I). That it may change entirely the appearance of the locus will be seen by a glance at figures 92, 93, and 94. The same curves are superposed in fig. 95. Though each locus may have its own double points, they cannot when superposed cut each other. 23. In the figures 96-103, the curves of the equation sin y sin fi/= — sm x sin ^x-\-k (16) are shown for certain specified values of k. The origin is the place of the isolated point in fig. 99. The several curves if superposed will not intersect. The values of k were selected so as to furnish curves with double points. 24. A series of twelve curves from the equation sin y sin x\y^ —sin x sm^jX-\-k (1 V) is given in the figures 104-115. By tracing any selected portions of the figure through the series the effect of the change in k will be seen. As in equation (16) values of k were chosen which give (except fig. 108) real double jDoints. In each case other curves of the series with real double points might have been given. 25. Another series of fourteen curves is given in figures 116-129 from the equation sin y sin ^y=sm x s'm^x-{-k. (18) The complete series would give 18 curves with double or isolated points. The omitted curves are those having isolated points, one at the beginning and one at the end of the series, one between figs. 127 and 128, and one between figs. 129 and 130. 26. Similar partial series can be seen in figs. 136-138, in figs. 139-143, and in figs. 144-146. 27. The superposition of the several curves of a series is shown in figure 147 where the curves represent the equation sin y sin f y= sin x sin fx-\-l: A little more than one complete figure of the curves is shown. The oi-igin is at the double point near the top of the figure. The value of k varies from curve to curve by intervals of -i^, and it cannot numeri- cally exceed 2, The full line corresponds to kz=0. The multiple that k is of -j-^ is denoted by the number of dots between the long strokes of the lines. The multiple that k is of — i is denoted by the number of short strokes between the long strokes in the lines. JVewtou and l*/ifl//ps on certain Transcendental Curves. lOT 28. The resemblance of lig. 147 to a series of contour lines in sur- veying, suggests a corresponding treatment of the equation. Let 2:=sin// sin/y?y— asina- ^mn.r—b (19) be the equation of a surface, and let it be intersected by planes parallel to the plane of .*-y, and we may obtain the groups of curves described in Arts. 22-27. The surface of equation (19) may be described by continuous mo- tion, as follows : Let sr=:siny sin my be a plane curve (figs. 1-37), and let it move parallel to itself so that each point of it shall describe a straight line parallel to the axis of .v. The curve shall then describe a cylindrical surface whose equation is zz=.%mi/ •t^mtny. (20) Let z=z — am\x^\\\nx — b be the equation of a second plane curve, and let this curve move parallel to the plane xz, in such a manner that the axis of x of the curve shall always lie in the cylindrical sur- face (20), The curve will describe by its motion the surface of equation (19). The surface will consist of one contini;ous sheet lying between the two parallel planes sr^it (l + ^-j-^'*), the positive numerical values of a and h being here taken. 29. By means of the two arbitrary constants, a and h, in equation (1) the curve may be made to pass through any two points of the plane. In a rectangle whose base is 2*7' ;r, and altitude IqTt, there are '^{p-\-q){p' -\-q') possible positions of double points (Art. 11, k.) If the curve passes through such a point it must have there two branches real or imaginary. Hence we may assign to a and b such values that the curve will have double points, in general, at any two of the ^{l^-\-q){l)'-{-q') possible positions. ERRATUM m PLATE XVI. In figure 40, plate XVI, there is a series of ovals about one-half of the real double points. There should be added to the curve, as represented, a like series of ovals around each of the remaining real double points. v. On the Equilibrium of Heterogeneous Substances. By J. WlLLARD GiBBS. "Die Energie der Welt ist constant. Die Entropie der Welt strebt elnem Maximum zu." Clausius.* The comprehension of the hiws which govern any material system is greatly facilitated by considering the energy and entropy of the system in the various states of which it is capable. As the difference of the values of the energy for any two states represents the com- bined amount of work and heat received or yielded by the system when it is brought from one state to the other, and the difference of entropy is the limit of all the possible values of the integral I -t'-i {dQ denoting the element of the heat received from external sources, and t the temperature of the part of the system receiving it,) the varying values of the energy and entropy characterize in all that is essential the effects producible by the system in passing from one state to another. For by mechanical and thermodynamic con- trivances, supposed theoretically perfect, any supply of work and heat may be transformed into any other which does not differ from it either in the amount of work and heat taken together or in the value of the integral / — —. But it is not only in respect to the extei'ual relations of a system that its energy and entropy are of predominant importance. As in the case of simply mechanical sys- tems, (such as are discussed in theoretical mechanics,) which are capable of only one kind of action upon external systems, viz., the perform- ance of mechanical work, the function which expresses the capability of the system for this kind of action also plays the leading part in the theory of equilibrium, the condition of equilibrium being that the variation of this function shall vanish, so in a thermodynamic system, (such as all material systems actually are,) which is capable of two different kinds of action upon external systems, the two functions which express the twofold capabilities of the system afford an almost equally simple criterion of equilibrium. *Pogg. Ann. Bd. cxxv (1865), S. 400; or Mechanische Warmetheorie, Abhand. ix., S. 44. J. W. Gihhs — Equilibrium of Heterogeneous Substances. 109 CRITERIA OP EQUlLIBRIUiM AND STABILITY. Tlie criterion of equilibrium for a material system Avhicli is isolated from all external influences may be expressed in either of the follow- ing entirely equivalent forms : I. M>r the equilibrium of any isolated si/stem it is necesmn/ and sufficient that in all possible variations of tlie state of the system which do not alter its energy^ the variation of its entropy shall either vanish or be negative. If e denote the energy, and ;/ the entropy of the system, and we use a subscript letter after a variation to indicate a quantity of which the value is not to be varied, the condition of equilibrium may be written {^V)e ^0- (1) II. For the equilibrium of any isolated system it is 7iecessary and sufficient that in cdl possible variations in the state of the system which do not alter its entropy^ the variation of its energy shall either vanish or be positive. This condition may be written (d>),^ 0. (2) That these two theorems are equivalent will appear from the con- sideration that it is always possible to increase both the energy and the entropy of the system, or to decrease both together, viz., by imparting heat to any part of the system or by taking it away. For, if condition (l) is not satisfied, there must be some variation in the state of the system for which 6i] > 0 and de =zQ; therefore, by diminishing both the energy and the entropy of the system in its varied state, we shall obtain a state for which (considered as a variation from the original state) 6i]z=i 0 and (^f <0; therefore condition (2) is not satisfied. Conversely, if condition (2) is not satisfied, there must be a variation in the state of the system for which (Jf < 0 and 6i]^^0\ hence there must also be one for which ^f rz: 0 and (J// > 0 ; therefore condition (1) is not satisfied. The equations which express the condition of equilibrium, as also its statement in words, are to be interpreted in accordance with the o-eneral usage in respect to differential equations, that is, infinitesimals 110 J.W. Gibbs — Equilihriiim of Heterogeneous Suhsta7ices. of higher orders than the first relatively to those which express the amount of change of the system are to be neglected. Biit to distin- guish the dilFerent kinds of equiliVjriam in respect to stability, we must have regard to the absolute values of the variations. We will use A as the sign of variation in those equations which are to be con- strued strictly, i. e., in which infinitesimals of the higher orders are not to be neglected. With this understanding, Ave may express the necessary and sufticient conditions of the difi:erent kinds of equi- librium as follows; — for stable equilibrium (^V)e<0, i.e., (A^),^>0: (3) for neutral equilibrium there must be some variations in the state of the system for which (A//)^:=rO, i. e., {A5)^^ = 0, (4) while in general (^V)e ^0, i.e., (A£)^^0; (5) and for unstable equilibrium tliere must be some variations for which (A;;),>0, (6) i. e., there must be some for which (^f),<0, " (V) while in general ((^;/),^0,i.e, (^6),^0. (8) In these criteria of equilibrium and stability, account is taken only oi possible variatic>ns. It is necessary to explain in what sense this is to be understood. In the first place, all variations in the state of the system which involve the transportation of any matter through any finite distance are of course to be excluded from consideration, although they may be capable of expression by infinitesimal varia- tions of quantities which perfectly determine the state of the system. For example, if the system contains two masses of the same sub- stance, not in contact, nor connected by other masses consisting of or containing the same substance or its components, an infinitesimal increase of the one mass with an equal decrease of the other is not to be considered as a possible variation in the state of the system. In addition to such cases of essential impossibility, if heat can pass by conduction or radiation from every j^art of the system to every other, only those variations are to be rejected as impossible, which involve changes which are prevented by passive forces or analogous resist- ances to change. But, if the system consist of parts between which there is supposed to be no thermal communication, it will be neces- sary to regard as impossible any diminution of the entropy of any of J. W. Gibbs — Equilibrium of Heterogeneous Sub.^taxces. 11] tliese parts, as sueli a change can not take place witliout the passage of heat. This limitation may most conveniently he applied to the second of the above forms of the condition of equilibrium, which will then become (^^V, ;/", etc. = t*. (9) ?/, //", etc., denoting the entropies of the various parts between which there is no communication of heat. When the condition of equi- librium is thus expressed, the limitation in respect to the conduction of heat will need no farther consideration. In order to apply to any system the criteria of equilibriiim which have been given, a knowledge is requisite of its passive forces or resistances to change, in so far, at least, as they are capable of pre- venting change. (Those passive forces which only retard change, like viscosity, need not be considered.) Such properties of a system are in general easily recognized upon the most superficial knowledge of its nature. As examples, we may instance the passive force of friction which prevents sliding when two surfaces of solids are pressed together, — that which prevents the different components of a solid, and sometimes of a fluid, from having different motions one from another, — that resistance to change which sometimes prevents either of two forms of the same substance (simple or compound), which are capable of existing, from passing into the other, — that which prevents the changes in solids which imply plasticity, (in other words, changes of the form to which tlie solid tends to return,) when the deformation does not exceed certain limits. It is a characteristic of all these passive resistances that they pre- vent a certain kind of motion or change, however the initial state of the system may be modified, and to whatever external agencies of force and heat it may be subjected, within limits, it may be, but yet within limits which allow finite variations in the values of all the quanti- ties which express the initial state of the system or the mechanical or thermal influences acting on it, without producing the change in question. The equilibrium which is due to such passive properties is thus widely distinguished from that caused by the balance of the active tendencies of the system, where an external influence, or a change in the initial state, infinitesimal in amount, is sufticient to pro- duce change either in the positi^-e or negative direction. Hence the ease with which these passive resistances are recognized. Only in the case that the state of the system lies so near the limit at which the resistances cease to be operative to prevent change, as to create a 112 tT. W. Gibbs — Equilibrium of Heterogeneous Substances. doubt whether the case falls within or witliout the limit, will a moi'e accurate knowledge of these resistances be necessary. To establisli the validity of the criterion of equilibrium, we will consider first the sufficiency, and afterwards the necessity, of the con- dition as expressed in either of the two equivalent forms. In the first place, if the system is in a state in which its entropy is greater than in any other state of the same energy, it is evidently in equilibrium, as any change of state must involve either a decrease of entropy or an increase ot energy, which are alike impossible for an iso- lated system.. We may add that this is a case of stable equilibrium, as no infinitely small cause (whether relating to a variation of the initial state or to the action of any external bodies) can produce a finite change of state, as this Avould involve a finite decrease of entropy or increase of energy. We will next suppose that the system has the greatest entropy consistent with its energy, and therefore the least energy consistent with its entropy, but that there are other states of the same energy and entropy as its actual state. In this case, it is impossible that any motion of masses should take place ; for if any of the energy of the system should come to consist of vis viva (of sensible motions), a state of the system identical in other respects but without the motion would have less energy and not less entropy, which would be contrary to the supposition. (But we cannot apply this reasoning J,o the motion within any mass of its different components in different directions, as in diffiision, when the momenta of the components balance one another.) Nor, in the case supposed, can any conduction of heat take place, for this involves an increase of entropy, as heat is only conducted from bodies of higher to those of lower temperature. It is equally impossible that any changes should be produced by the transfer of heat by radiation. The condition which we have sup- posed is therefore sufficient for equilibrium, so far as the motion of masses and the transfer of heat are concerned, but to show that the same is true in regard to the motions of diffusion and chemical or molecular changes, when these can occur without being accompanied or followed by the motions of masses or the transfer of heat, we must have recourse to considerations of a more general nature. The fol- lowing considerations seem to justify the belief that the condition is sufficient for equilibrium in every respect. Let us suppose, in order to test the tenability of such a hypothesis, that a system may have the greatest entropy consistent with its energy without being in equilibrium. In such a case, changes in the J, W. Gihhs — Equilihrkmi of Heterogeneous Substances. 113 state of the system must take place, but these will necessarily be such that the energy and the entropy will remain unchanged and the system will continue to satisfy the same condition, as initially, of having the greatest entropy consistent with its energy. Let us con- sider the change which takes place in any time so short that the change may be regarded as uniform in nature throughout that time. This time must be so chosen that the change does not take place in it infinitely slowly, which is always easy, as the change which we sup- pose to take place cannot be infinitely slow except at particular moments. Now no change whatever in the state of the system, which does not alter the value of the energy, and which commences with the same state in which the system was supposed at the com- mencement of the short time considered, will cause an increase of entropy. Hence, it Avill generally be possible by some slight varia- tion in the circumstances of the case to make all changes in the state of the system like or nearly like that which is supposed actually to occur, and not involving a change of energy, to involve a necessary decrease of entropy, which would render any such change impossible. This variation may be in the values of the variables which determine the state of the system, or in the values of the constants which deter- mine the nature of the system, or in the form of the functions which express its laws, — only there must be nothing in the system as modi- fied which is thermodynamically impossible. For example, we might suppose teraperatiire or pressure to be varied, or the composition of the diiFerent bodies in the system, or, if no small variations which could be actually realized would produce the required result, we might suppose the properties themselves of the substances to undergo variation, subject to the general laws of matter. If, then, there is any tendency toward change in the system as first supposed, it is a tendency which can be entirely checked by an infinitesimal variation in the circumstances of the case. As this supposition cannot be allowed, we must believe that a system is always in equilibrium when it has the greatest entropy consistent with its energy, or, in other words, when it has the least energy consistent with its entropy. The same considerations will evidently apply to any case in which a system is in such a state that A;? ^ 0 for any possible infinites- imal variation of the state for which Ae= 0, even if the entropy is not the least of which the system is capable with the same energy. (The term possible has here the meaning previously defined, and the character A is used, as before, to denote that the equations are to be Trans. Conn. Acad., Vol. III. 15 October, 1875. 114 '/. W. Gibhs — Equilibrium of Heterogeneous Substances. construed strictly, i. e., without neglect of the infinitesimals of the higher orders.) The only case in which the sufficiency of the condition of equi- librium which has been given remains to be proved is that in which in our notation dj] ^ 0 for all possible variations not affecting the energy, but for some of these variations A// > 0, that is, when the entroj^y has in some respects the characteristics of a minimum. In this case the considerations adduced in the last paragraph will not apply without modification, as the change of state may be infinitely slow at first, and it is only in the initial state that the condition Sr^^ -S 0 holds true. But the differential coefficients of all orders of the quantities which determine the state of the system, taken with respect of the time, must be functions of these same quantities. None of these differential coefficients can have any value other than 0, for the state of the system for which 8ri^ ^0. For otherwise, as it would generally be possible, as before, by some infinitely small modification of the case, to render impossible any change like or nearly like that which might be supposed to occur, this infinitely small modification of the case would make a finite difference in the value of the differential coefficients which had before the finite values, or in some of lower orders, which is contrary to that continuity which we have reason to expect. Such considerations seem to justify us in regarding such a state as we are discussing as one of theoretical equilibrium ; although as the equilibrium is evidently unstable, it cannot be realized. We have still to prove that the condition enunciated is in every case necessary for equilibrium. It is evidently so in all cases in which the active tendencies of the system are so balanced that changes of every kind, except those excluded in the statement of the condition of equilibrium, can take place reversibly, (i. e., both in the positive and the negative direction,) in states of the system dif- fering infinitely little from the state in question. In this case, we may omit the sign of inequality and write as the condition of such a state of equilibrium (0»,rr:0, i.e., {6e\ = 0 (10) But to prove that the condition previously enunciated is in every case necessary, it must be shown that whenever an isolated system remains without change, if there is any infinitesimal variation in its state, not involving a finite change of position of any (even an infini- tesimal part) of its matter, which would diminish its energy by a J. ir. Gihhs — Equilibrium of Heterogeneous Substances. 115 quantity which is not infinitely small relatiA^ely to the variations of the quantities which determine tlie state of the system, without altering its entropy,— or, if the system has thermally isolated parts, without altering the entropy of any such part, — this variation involves changes in the system which are prevented by its passive forces or analogous resistances to change. Now, as the described variation in the state of the system diminishes its energy without altering its entropy, it must be regarded as theoretically possible to produce that variation by some process, perhaps a very indirect one, so as to gain a certain amount of work (above all expended on the system). Hence we may conclude that the active forces or tenden- cies of the system favor the variation in question, and that equi- librium cannot subsist unless the variation is prevented by passive forces. The preceding considerations will suffice, it is believed, to establish the validity of the criterion of equilibrium which has been given. The criteria of stability may readily be deduced from that of equi- librium. We will now proceed to apjily these principles to systems consisting of heterogeneous substances and deduce the special laws which apply to different classes of phenomena. For this purpose we shall use the second form of the criterion of equilibrium, both because it admits more readily the introduction of the condition that there shall be no thermal communication between the different parts of the system, and because it is more convenient, as respects the form of the general equations relating to equilibrium, to make the entropy one of the independent variables which determine the state of the system, than to make the energy one of these variables. THE CONDITIONS OF EQUILIBRIUiNt FOR HETEROGENEOUS MASSES IN CONTACT WHEN UNIXFLtTENCED BY GRAVITY, ELECTRICITY, DISTORTION OF THE SOLID MASSES, OR CAPILLARY TENSIONS. In order to arrive as directly as })ossible at the most characteristic and essential laws of chemical equilibrium, we will first give our attention to a case of the simplest kind. We will examine the con- ditions of equilibrium of a mass of matter of various kinds enclosed in a rigid and fixed envelop, which is impermeable to and unalter- able by any of the substances enclosed, and perfectly non-conducting to heat. We will suppose that the case is not complicated by the action of gravity, or by any electrical influences, and that in the solid portions of the mass the pressure is the same in every direction. 116 J. W. Glbbs — Equilihrium of Heterogeneous Substances. We will farther simplify the problem by supposing that the varia- tions of the parts of the energy and entropy which depend upon the surfaces separating heterogeneous masses are so small in comparison with the variations of the parts of the energy and entropy which depend upon the quantities of these masses, that the former may be neglected by the side of the latter; in other words, we will exclude the considerations which belong to the theory of capillarity. It will be observed that the siipposition of a rigid and non- conducting envelop enclosing the mass under discussion involves no real loss of genei-ality, for if any mass of matter is in equilibrium, it would also be so, if the whole or any part of it were enclosed in an envelop as supposed ; therefore the conditions of equilibrium for a mass thus enclosed are the general conditions which must always be satisfied in case of equilibrium. As for the other suppositions which have been made, all the circumstances and considerations which are here excluded will afterward be made the subject of special discussion. Conditions relating to the Equilibrium between the initially existing Hoinogeneons Partt^ of the given Mass. Let us first consider the energy of any homogeneous part of the given mass, and its variation for any j^ossible variation in the com- position and state of this part. (By homogeneous is meant that the part in question is uniform throughout, not only in chemical com- position, but also in physical state.) If we consider the amount and kind of matter in this homogeneous mass as fixed, its energy 5 is a function of its entropy ?/, and its volume v, and the differentials of these quantities are subject to the relation ds. ■=. t di] - • p dv ., (11) t denoting the (absolute) temperature of the mass, and p its pressure. For t di] is the heat received, and p do the work done, by the mass during its change of state. But if we consider the matter in the mass as variable, and write mj, jn^, . . . m„ for the quantities of the various substances /S'j, /Sg, . . . N„ of which the mass is composed, s will evidently be a function of //, v, m^., ^2, . . . ?>?„, and we shall have for the complete value of the differential of e de:=ztdi] — pdv -{- f.i^dm^-\- I.i.,dm2 . . . -|-//„(?ot„, (12) yUj, yWg, . . . //„ denoting the diflferential coefficients of s taken with respect to m,, nio, . . . m„. The substances /S',, 62, . . . /S'„, of which we consider the mass composed, must of course be such that the values of the differen- J. W. Gibbs — Equilibrium of Heterogeneous Substances. Il7 tials c?mj, dm^., . . . dm^ shall be indeiDendent, and shall express every possible variation in the composition of the homogeneous mass considered, including those produced by the absorption of substances different from any initially pi-esent. It may therefore be necessary to have terms in the equation relating to component substances which do not initially occur in the homogeneous mass considered, provided, of course, that these substances, or their components, are to be found in some part of the whole given mass. If the conditions mentioned are satisfied, the choice of the sub- stances which we are to i-egard as the components of the mass con- sidered, may be determined entirely by convenience, and independently of any theory in regard to the internal constitution of the mass. The number of components will sometimes be greater, and sometimes less, than the number of chemical elements present. For example, in considering the equilibrium in a vessel containing water and free hydrogen and oxygen, we should be obliged to recognize three com- ponents in the gaseous part. But in considering the equilibrium of dihite sulphuric acid with the vapor which it yields, we shoiild have only two components to consider in the liquid mass, sulphuric acid (anhydrous, or of any particular degree of concentration) and (addi- tional) water. If, however, we are considering sulphuric acid in a state of maximum concentration in connection with substances which might possibly afford water to the acid, it must be noticed that the condition of the independence of the differentials will require that we consider the acid in the state of maximum concentration as one of the components. The quantity of this component will then be capa- ble of variation both in the positive and in the negative sense, while the quantity of the other component can increase but cannot decrease below the value 0. For brevity's sake, we may call a substance S^ an actual component of any homogeneous mass, to denote that the quantity ra^ of that substance in the given mass may be either increased or diminished (although we may have so chosen the other component substances that m^ =. 0) ; and we may call a substance S^ a possible component to denote that it may be combined with, but cannot be substracted from the homogeneous mass in question. In this case, as we have seen in the above example, we must so choose the component sub- stances that Wj rz 0. The units by which we measure the substances of which we regard the given mass as composed may each be chosen independently. To fix our ideas for the purpose of a general discussion, we may suppose 118 J. W. Gihhs — Equilibrium of Heterogeneous Substances. all substances measured by weight oi' mass. Yet in special cases, it may be more convenient to adopt chemical equivalents as the units of the component substances. It may be observed that it is not necessary for the validity of equation (12) that the variations of nature and state of the mass to which the equation refers should be such as do not disturb its homo- geneity, provided that in all parts of the mass the variations of nature and state are infinitely small. For, if this last condition be not violated, an equation like (12) is certainly valid for all the infin- itesimal parts of the (initially) homogeneous mass; i. e., if we write 2>f, Z>//, etc., for the energy, entropy, etc., of any infinitesimal part, dDe = t dDt] - p dBv -\- /< ^ dDm ^-\- fi.^ dJDni^ ...-{-/'« dDm„^ (13) whence we may derive equation (12) by integrating for the whole initially homogeneous mass. We will now suppose that the whole mass is divided into parts so that each part is homogeneous, and consider such variations in the energy of the system as are due to variations in the composition and state of the several parts i*emaining (at least approximately) homoge- neous, and together occupying the whole space within the envelop. We will at first suppose the case to be such that the component sub- stances are the same for each of the parts, each of the substances aSj, *S'2, . . . Sn being an actual component of each part. If we distinguish the letters referring to the different parts by accents, the variation in the energy of the system may be expressed by Se' -\- Se" -\- etc., and the general condition of equilibrium requires that (Jt'+.f^f" -h etc, ^ 0 (14) for all variations which do not conflict with the equations of condi- tion. These equations must express that the entropy of the whole given mass does not vary, nor its volume, nor the total quantities of any of the substances ^Sj, aS^j, . . . Sn- We will suppose that there are no other equations of condition. It will then be necessary for equilibrium that i; 8r,' ^ p' 6v' H-///(Jm,' -\- i.i„' dm^J . J^t" 67}" - p" 6v" -\- 1.1 ^" dm ^" + lA^" dm.J' . -I- etc. ^ 0 for any values of the variations for which 6if -f 67]" + 67/" 4- etc. = 0, Sv' -j- 6v" + 6v"' + etc. = 0, + /'„' 6m.: + yw„" 6m„ II (15) (16) (17) J. W. Gibbs — JEquilihrium of Heterogeneous Substances. 119 (18) drn^' -\- dm/' + dm/" + etc. = 0, ] dm./ + dm/' + dm/" + etc. =: 0, and dm/ -\- dm/' -j- dm/" -\- etc. = 0. For this it is evidently necessary and sufficient that t' = t" =zt"'z:i etc. (19) y =y =y' — etc. (20) /Yj' = //,"=///"= etc.^ f.i/ — H/' z= ^i/" = etc. [ ^21) lA,! z= pi/' = fx/" =. etc. J Equations (19) and (20) express the conditions of thermal and mechanical equilibrium, viz., that the temperature and the pressure must be constant throughout the whole mass. In equations (21) we have the conditions characteristic of chemical equilibrium. If we call a quantity //„ as defined by such an equation as (12), the potential for the substance >S, in the homogeneous mass considered, these con- ditions may be expressed as follows : The potential for each cotnponent substance must be constant throughout the lohole mass. It will be remembered that we have supposed that there is no restriction upon the freedom of motion or combination of the com- ponent substances, and that each is an actual component of all parts of the given mass. The state of the whole mass will be completely determined (if we regard as immaterial the position and form of the various homoge- neous parts of which it is composed), when the values are determined of the quautities of whicli the variations occur in (15). The number of these quantities, which we may call the independent variables, is evidently {n -\- 2) k, k denoting the number of homogeneous parts into which the whole mass is divided. All the quantities which occur in (19), (20), (21), are functions of these variables, and may be regarded as known functions, if the energy of each part is known as a function of its entropy, volume, and the quantities of its com- ponents. (See eq. (12).) Therefore, equations (19), (20), (21), may be regarded as {v - 1) {n -\- 2) independent equations between the independent variables. The volume of the whole mass and the total quantities of the various substances being known afford n-\- \ addi- tional equations. If we also know the total energy of the given mass, or its total entropy, we will have as many equations as there are independent variables. 120 J^ W. Gibbs — Equilibrmm of Heterogeneous Substances. But if any of the substances S^, S.^ . . . S„ are only possible com- ponents of some parts of the given mass, the variation 6m of the quantity of such a substance in such a part cannot have a negative value, so that the general condition of equilibrium (15) does not require that the potential for that substance in that part should be equal to the potential for the same substance in the parts of which it is an actual component, but only that it shall not be less. In this case instead of (21) we may write for all parts of which ^S'j is an actual component, and for all parts of which S^ is a possible (but not actual) com- i ponent, ' Ih = ^^2 y (22) for all parts of which iS'g is an actual component, and for all parts of which S2 is a possible (but not actual) com- ponent, etc., J/j, M2, etc., denoting constants of which the value is only deter- mined by these equations. If we now suppose that the components (actual or possible) of the various homogeneous parts of the given mass are not the same, the result will be of the same character as before, provided that all the different components are indej^endeyit, (i. e., that no one can be made out of the others,) so that the total quantity of each component is fixed. The general condition of equilibi'ium (15) and the equations of condition (16), (17), (18) Avill require no change, except that, if any of the substances S^ , S2 . . . S„ is not a component (actual or possible) of any part, the term fx dm for that substance and part will be wanting in the former, and the 6m in the latter. This will require no change in the form of the particular conditions of equilibrium as expressed by (19), (20), (22); but the number of single conditions contained in' (22) is of course less than if all the component sub- stances were components of all the parts Whenever, therefore, each of the different homogeneous parts of the given mass may be regarded as composed of some or of all of the same set of substances, no one of which can be formed out of the others, the condition which (with equality of temperature and pressure] is necessary and sufficient for equilibrium between the different parts of the given mass may be expressed as follows: J. W. Gihbs — Equilihrmm of Heterogeneous Substances. 121 The potentUd for each of the component substances must luioe a constant value in all parts of the given mass of iddch that substance is an actual conxponeyit^ and have a value not less than this in all parts of which it is a possible componetit. The number of equations aftbrded by these conditions, after elimina- tion of J/j, iT/g, . . . Jf„, will be less than {n +• 2) (k - 1) by the num- ber of terms in (15) in which the variation of the form dm is either necessarily nothing or incapable of a negative value. The number of variables to be determined is diminished by the same number, or, if we choose, Ave may write an equaticm of the form m — 0 for each of these terms. But when the substance is a possible component of the part concerned, there will also be a condition (expressed by ^ ) to show whether the supposition that the substance is not an actual component is consistent with equilibrium. We will now suppose that the substances S-^^, 8^, . . . iS„ are not all independent of each other, i. e., that some of them can be formed out of others. We will first consider a very simple case. Let S^ be composed of S^ and So combined in the ratio of a. to b, S^ and S2 occurring as actual components in some parts of the given mass, and /S'g in other parts, which do not contain S^ and S2 as separately A^ariable components. The general condition of equilibrium will still have the form of (15) with certain of the terms of the form /< dm omitted. It may be written more briefly [(23) ^{tSi/) - 2{pdv)-^:::^{/.i,(hn^)-^2{/'2dm2) ■ ' .-\-^^{Mn<^'n„)^0, the sign ^ denoting suumiation in regard to the difierent parts of the given mass. But instead of the three equations of condition, 2 6m 1=0, 2" dm2 = 0, 2 6m^ — 0, (24) we shall have the two, 2Sm,+^^2Sm, = 0,] The other equations of condition, 2 Sij = 0, :2 dv = 0, ^ Sm^ = 0, etc., (26) will remain unchanged. Now as all values of the variations which satisfy equations (24) will also satisfy equations (25), it is evident that all the particular conditions of equilibrium which we have already deduced, (19), (20), (22), are necessary in this case also. When these are satisfied, the general condition (23) reduces to M, 2 6)n , -f 3f, 2 6m 2+ M^ 2 6m 3^0. (27) Trans. Conn. Acad. 16 October. 1875. ;. (25) 122 J. IF. Glbhs — Equilihrium of Heterogeneous Substances. For, although it may be that //j', for example, is greater than J/^, yet it can only be so when the following Sin^' is incapable of a nega- tive value. Hence, if (27) is satisfied, (23) must also be. Again, if (23) is satisfied, (27) must also be satisfied, so long as the variation of the quantity of every substance has the value 0 in all the parts of which it is not an actual component. But as this limitation does not affect the range of the possible values of 2 6m ^, 2 dni^, and 2E Sm^, it may be disregarded. Therefore the conditions (23) and (27) are entirely equivalent, when (19), (20), (22) are satisfied. Now, by means of the equations of condition (25), we may eliminate 2 6m^ and ^6)712 from (27), which becomes - a 31 ^ 2 6m ^ — hM^ 2 6m^ + {a + h) M^:S 6m ^ ^ 0, (28) i.e., as the value of 2 6m^ may be either positive or negative, a M^ ■\-bM2 — {a-\-h) M^, (29) which is the additional condition of equilibrium which is necessary in this case. The relations between the component substances may be less simple than in this case, but in any case they will only affect the equations of condition, and these may always be found without diffi- culty, and will enable us to eliminate from the general condition of equilibrivim as many variations as there are equations of condition, after which the coefficients of the remaining variations may be set equal to zero, except the coefficients of variations which are incapable of negative values, which coefficients must be equal to or greater than zero. It will be easy to perform these operations in each par- ticular case, but it may be interesting to see the form of the resultant equations in general. We will suppose that the various homogeneous parts are considered as having in all n comjjonents, «Sj, aS'^, . . . iS„, and that there is no restriction upon their freedom of motion and combination. But we Avill so far limit the generality of the problem as to suppose that each of these components is an actual component of some part of the given mass.* If some of these components can be formed out of others, all such relations can be expressed by equations such as ^^ ©a + P S/, + etc. = n e, 4- A i5, + etc. (30) where ©a, <Si, ®a, etc. denote the units of the substances /S„, Si,, S^, etc., * When we come to seek the conditions of equilibrium relating to the formation of masses unlike any previously existing, we shall take up de novo the whole problem of the equilibrium of heterogeneous masses enclosed in a non-conducting envelop, and give it a more general treatment, which will be free from this limitation. J. W. Gibbs —Equilibrium of Heterogeneous Substances. 123 (that is, of certain of tlie substances »S'j, ^S'^, . . . N,,,) and «-, /:/, h etc. denote numbers. These are not, it will be observed, equations between abstract quantities, but the sign =z denotes qualitative as well as quantitative equivalence. We will suppose that there are r independent equations of this character. The equations of con- dition relating to the component substances may easily be derived from these equations, but it will not be necessary to consider them particularly. It is evident that they will be satisfied by any values of the variations which satisfy equations (18); hence, the particular conditions of equilibrium (19), (20), (22) must be necessary in this case, and, if these are satisfied, the general equation of equilibrium (15) or (2.3) will reduce to J/, >; dm 1 + J/g :^ drii^ . . . -}- 31^2 6m„^ 0. (31) This will appear from the same considerations which were used in regard to equations (2.3) and (27). Now it is evidently possible to give to 2 Sm^, 2 dm,„ 2 Snii., etc. values proportional to a, fi, — ;<:, etc. in equation (-30), and also the same values taken negatively, making 2 dm =^ 0 in each of the other terms ; therefore aM^ + pM,-\- etc. . . . - « J/^. -XM,^ etc. ::^ 0, (32) or, a M„ -\- f-i M,, + etc. = u M^ -\- X 31^ + etc. (33) It will be observed that this equation has the same form and coeifi- cients as equation (30), JI taking the place of ©. It is evident that there must be a similar condition of equilibrium for every one of the r equations of which (30) is an example, which may be obtained sim- ply by changing © in these equations into 3f, When these condi- tions are satisfied, (31) will be satisfied with any possible values of 2 6m I, 2 Sni^, , . . 2 drii^. For no values of these quantities are possible, except such that the equation {2dm,)(S,-^{2dm.,)(B2 . . . -\-{2dm,)e„=0 (84) after the substitution of these values, can be derived from the r equa- tions like (30), by the ordinary processes of the reduction of linear equations. Therefore, on account of the correspondence between (31) and (34), and between the r equations like (33) and the r equations like (30), the conditions obtained by giving any possible values to the variations in (31) may also be derived from the r equations like (33) ; that is, the condition (31) is satisfied, if the r equations like (33) are satisfied. Therefore the r equations like (33) are with (19), (20), and (22) the equivalent of the general condition (15) or (23). 124 J. W. Gibbs — Equilibrimn of Heterogeneous ^Substances. For determining the state of a given mass when in equilibrium and having a given vohime and given energy or entropy, the condi- tion of equilibrium affords an additional equation corresponding to each of the r independent relations between the n component sub- stances. But the equations which express our knowledge of the matter in the given mass will be correspondingly diminished, being n — r m number, like the equations of condition relating to the quantities of the component substances, which may be derived from the former by differentiation. Conditions relating to the possible formation of Masses Unlike any Preiiiousli/ Existing. The variations which we have hitherto considered do not embrace every possible infinitesimal variation in the state of the given mass, so that the particular conditions already formed, although always necessary for equilibrium (when there are no other equations of con- dition than such as we have supposed), are not always sufficient. For, besides the infinitesimal variations in the state and composition of different parts of the given mass, infinitesimal masses may be formed entirely different in state and composition from any initially existing. Such parts of the whole mass in its varied state as cannot be regarded as parts of the initially existing mass which have been infinitesimally varied in state and composition, we will call ne^o parts. These will necessai'ily be infinitely small. As it is more convenient to regard a vacuum as a limiting case of extreme rarefaction than to give a special consideration to the possible for- mation of empty spaces within the given mass, the term new parts Avill be used to include any empty spaces which may be formed, when such have not existed initially. We will use De, D?], Dv, Dm^, X>w?2, . . . Din^ to denote the infinitesimal enei'gy, entropy, and vol- ume of any one of these new parts, and the infinitesimal quantities of its components. The component substances 8^,8^,. . . S„ must now be taken to include not only the independently variable com- ponents (actual or possible) of all parts of the given mass as initially existing, but also the components of all the new parts, the possible formation of which we have to consider. The character S will be used as before to express the infinitesimal variations of the quantities relating to those parts which are only infinitesimally varied in state and compc^sition, and which for distinction we will call orif/inal parts, including under this term the empty sj^aces, if such exist initially, within the envelop bounding the system. As we may divide the given mass into as many parts as we choose, and as not only the J. W. Glhhs — Equilibrium of Heterogeneous Substances. 125 initial bounclarie!^, but also the movements of these bomidaries during any variation in the state of the system are arbitrary, we may so define the parts which we have called original, that we may consider them as initially homogeneous and remaining so, and as initially con- stituting the whole system. The most general value of the energy of the whole system is evidently ^68-\-^J)^, (35) the first summation relating to all the original parts, and the second to all the new parts. (Throughout the discussion of this problem, the letter 6 or D following ^ will sufficiently indicate whether the sum- mation relates to the original or to the new parts.) Therefore the general condition of equilibrium is :^de-it- :^6e^0, (36) or, if w^e substitute the value of de taken from equation (12), [(37) ^De^^{tSii) - 2{2>dv)-\-2{i.i^dm^)-{.:£{iJ^6m.,) . . +^^(/v?w„)^ 0. If any of the substances S ^^ S^^ . . . *S'„ can be formed out of others, we will suppose, as before (see page 122), that such relations are expressed by equations betw^een the units of the different substances. Let these be «j ®1 -f «2 ®2 • • • + ^nSn^ 0 j ^1 ®i + ^''s ®3 • • • + ''■'n ®n = 0 >• ^equations, (38) etc. ) The equations of condition will be (if there is no restriction upon the freedom of motion and composition of the components) :E6t] + ^D)]=Q, (39) :E6v -\-2I>V:=iO, (40) and n — r equations of the form + h„ (:S' 8m„ + '2 Dm.„) = 0 |' ^^ {2 6m^ +2 Dm,) + z, (2 Sm., + 2 Dm„) . . ^ (41)* + /„ {2 Sm„ + 2 Dm„) = 0 etc. * In regard to the relation between the coefficients in (41) and those in (38), the reader will easily convince himself that the coefficients of any one of equations (41) are such as would satisfy all the equations (38) if substituted for Sj, .S'^, . . . S„; and that this is the only condition which these coefficients must satisfy, except that the .fi _ r sets of coefficients shall be independent, i. e., shall be such as to form inde- pendent equations ; and that this relation between the coefficients of the two sets of equations is a reciprocal one. 126 J. TT. (rihh$ — Equilibrium of ffetero(jfefieous Substances. Xow, using Lagrange's ''■method of multipliers,"* we will sul)- tract 7' {:^ 6rf + I^ Dr>) -P(2:'o\' -\- 2: I)r) from the first member of the general condition of equilibrium (^H), 7' and P being constants of which the value is as vet arbitrary. We might proceed in the same way with the remaining equations of condition, but we may obtain the same result more simply in another way. We will first observe that + (:i" 6m„ + >: Dm„) S„ = 0, (42) which equation would hold identically for any possible values of the quantities in the parentheses, if for r of the letters 3j, 3^, . . . ^„ were substituted their values in terms of the others as derived from equa- tions (38). (Although 2 ,, Sg^ . . . 3n do not represent abstract quanti- ties, yet the operations necessary for the reduction of linear equations are evidently applicable to eqiuitious (38).) Therefore, equation (42) will hold true if for 3^, Sg, . . . 2„ we substitute n numbers which satisfy equations (38). Let 3/,, J/j, . . . 3I„ be such numbers, i. e., let ^»j J/j + bo 3I2 . . . + b^ J/„ = 0, '^ r equations, (43) etc. ) then J/j {:^Sm^^ :^Dm^) + M2 {2 6m2-h2Dm2) . . . + J/„ {:S 6m„ + 2i' Din„) = 0. (44) This expression, in which the values of « — r of the constants J/,, J/g? . . . JI„ are still arbitrary, we will also subtract from the first mem- ber of the general condition of equilibrium (37), which will then become 2D€+ 2 {t d>;) - :^ (/) dv) -{- 2 (;/ ,6m,) . . + 2: (//„ 6m„) - T2 d// + 1^2 6v - M, 2 6m , . . . + J/'„ v (^m„ - T2Dr^-\-F:SDv -3/, :2Dm, .. . -J/„ >Z)w„^0. (45) That is, having assigned to T, P, Jl,, JJ^, . . . 3/„ any values con- sistent with (43), we may assert that it is necessaiy and sufficient for equilibrium that (45) shall hold true for any variations in the state of the system consistent with the equations of condition (39), (40), (41). But it will always be possible, in case of equilibrium, to assign such values to T, P, M,^ Jf^, . . M^, without violating equations (43), * On account of the sign ^ in (37), and because some of the variations are incapable of negative values, the successive steps in the reasoning vriU be developed at greater length than would be otherwise necessary. ,1. \V. Oihbs — Kijidlihriidii of n>fii'i><inivi>i(H Su/isfa )/<•,. t. 127 tli:it (4">) shall lioM Inic tor all va^iati^•n^ in ihc slate of the system ami ill tlie <]uaiititi('s ot' tlie various Hulistanees eomposiiitj it, even tlioui;;li these v;iriati<>iis an- not consistent with the e(|nations of con- dition (39), (40), (41), For, when it is not |>ossil)le \i, dn this, it must be possil)le by a|i|)lyin«; (45) to variations in the HyKtcrn not necessarily restricletl by the etjuations of condition (-M*), (40), (41) to obtain conditions in re«:;ard to 7\ /\ .l/,, .1/,,, . . . M„, Home of which will be inconsistent with others or with c(|nalions (4:i). 'I'liese conditions we will repreBoiit by -1=0, 7?^ 0, etc., (JC.) .1, />, etc. beint; lineai- functions of 7', /', .1/,, .1/.,, . . .1/,. Then it will be possible to deduce fioni these conditions a sinj^le condition of the ft)rin n A + /i n + etc. ^0, (47) (K, fi, v\c. beini; positive constants, which cannot hold true consist- ently with ecpuvtions (43). Hut it is evident from the form of (47) tliat, like any of the eontlitions (40), it couhl have been obtained directly from (4')) by applying this formula to a certain chanj.(e in tlie system (|)erha])S not restrictcil by the ecpiations ot condition (30), (40), (41)). Now as (47) cannot hold true consistently with eqs. (43), it is evident, in the tirst place, that it cannot contain 7'or /*, there- fore in the diange in the system just mentioneil (for which (45) reduces to (47)) 2^6>/ + ::: J>/f=^\ and 2: O/- -f 2:' />>/" = 0, so that the equations of condition (39) aiid (40) are satisfied. Again, for the same reason, the homogeneous function of the first degree of JAj, J/o, . . . J/„ in (47) must be one of which the value is fixed by eqs. (43). l)ut the value thus fixed can only be zero, as is evident from the form of these equations. Therefore ( >■ ()■/// , + :^' it/n , ) J/, + ( :i" 6)n , + ::i" Um^ ) j/, . . . + ( ^" (h,}„ + :i lJni„) J/„ — 0 (4 8) for any values of ^/^, M^ . . . J/„ which satisfy eqs. (43), and theretore (I^'fJ///, + >Z>///i) 3j+ (:^'(J;«2 + ^"^'"2) 2j • • • + ( >' d'i/,„ + >■ Jjjn„) 2„ = 0 (49) for any numerical values of 2^^, Z2, . • . 3„ wdiich satisfy e«is, (3ft). This equation (40) will therefore hold true, if for r of the letters 3,2.,.. 3„ we substitute their values in terms of the others taken from eqs. (38), and therefore it will hold true when we use 128 J, W. Gibhs — Equilibrium of Heterogeneous Substances. <Sj, ®2, • • • ©n, as before, to denote the units of the various com- ponents. Thus understood, the equation expresses that the vahies of the quantities in the parentheses are such as are consistent with the equations of condition (41). The change in the system, there- fore, which we are considering, is not one which violates any of the equations of condition, and as (45) does not hold true for this change, and for all values of 2\ P, J/^, J/2, • • • ^^A which are consistent with eqs. (43), the state of the system cannot be one of equilibrium. Therefore it is necessary, and it is evidently sufficient for equilibrium, that it shall be possible to assign to 2\ P, 31^, 31^, . . . M„ such values, consistent with eqs. (43), that the condition (45) shall hold true for any change in the system irrespective of the equations of condition (39), (40), (41). For this it is necessary and sufficient that t=2\ p = P, (50) for each of the original parts as previously defined, and that Be - TDi]^PDr - J/, X*;//, - M^Dm^ ... - 3I,,Dm„^ 0, (52) for each of the 7iew parts as previously defined. If to these condi- tions we add equations (43), we may treat 1\ P, J/^, J/g, . . . J/„ simply as unknown quantities to be eliminated. In regard to conditions (51), it will be observed that if a sub- stance, 6' J, is an actual component of the part of the given mass distinguished by a single ac<^ent, dtn^' may be either positive or negative, and we shall have fi^' = M^ ; but if S^ is only a possible component of that part, (Sm^' will be incapable of a negative value, and we will have /^^'^ M^. The formula^ (S*^)- (51), ^^^^^ (43) express the same particular con- ditions of equilibrium which we have before obtained by a less gen- eral process. It remains to discuss (52). This formula must hold true of any infinitesimal mass in the system in its varied state which is not approximately homogeneous with any of the surrounding masses, the expressions i>£, J9//, i>y. Dm ^, J)in2, . . . Dm„ denoting the energy, entropy, and volume of this infinitesimal mass, and the quantities of the substances S^, S.y, . . . *S'„ which we regard as com- posing it, (not necessarily as independently variable components). If there is more tlian one way in which this mass may be considered as composed of these substances, we may choose whichever is most convenient. Indeed it follows directly from the relations existing between J/j, J/g, . . . and J/„ that the result Avould be the same in ./ W. Gibbs — Equilibriuiu of Heterogeneous /Substances. rJ9 any case. Now, if we assume tliat the values of i>f, />;/, JDo, Drn^, Dm.^, . . . Din„ are proportional to the values of f, //, v, m,, jh.^, . . . m„ for any large homogeneous mass of similar composition, and of the same temperature and pressure, the condition is equivalent to this, that € - T?^ + Pv - 3Ij m^ -3I2 in^ ... - iT/„m„ ^ 0 (53) for any large homogeneous body which can be formed out of the substances aS'j, S2 . . . S„. But the validity of this last transformation cannot be admitted without considerable limitation. It is assumed that the relation between the energy, entropy, volume, and the quantities of the dif- ferent components of a very small mass surrounded by substances of diiferent composition and state is the same as if the mass in ques- tion formed a jaart of a large homogeneous body. We started, indeed, with the assumption that we might neglect the part of the energy, etc., depending upon the surfaces separating heterogeneous masses. Now, in many cases, and for many purposes, as, in general, when the masses are large, such an assumption is quite legitimate, but in the case of these masses which are formed within or among substances of different nature or state, and which at their first forma- tion mi;st be infinitely small, the same assumption is evidently entirely inadmissible, as the siirfaces must be regarded as infinitely large in proportion to the masses. We shall see hereafter what modifications are necessary in our formula in order to include the parts of the energy, etc., which are due to the surfaces, but this will be on the assinnption, which is usual in the theory of capillarity, that the radius of curvature of the surfaces is large in proportion to the radius of sensible molecular action, and also to the thickness of the lamina of matter at the surface which is not (sensibly) homoge- neous in all respects with either of the masses which it separates. But although the formula? thus modified will apply with sensible accuracy to masses (occurring within masses of a diftei'ent nature) much smaller than if the terms relating to the surfaces were omitted, yet their failure when applied to masses infinitely small in all their dimensions is not less absolute. Considerations like the foregoing might render doubtful the validity even of (52) as the necessary and sufiicient condition of equilibrium in regard to the formation of masses not approximately homogeneous with those previously existing, when the conditions of equilibrium between the latter are satisfied, unless it is shown that in establishing this formula there have been no quantities neglected relating to the Trans. Conn. Acad., Vol. III. 17 October, 1875. 130 ./. IK (xibbs — EnullibrlaiH of Heteroaeneous Substances. mutual actiou of the new and the original parts, which can aftect the result. It will be easy to give such a meaning to the expressions 7>f, Di}, DiJ, D„i^, Dni^, . . . Dm„ that this shall be evidently the case. It will be observed that the quantities represented by these expressions have not been pei-fectly defined. In the first place, we have no right to assume the existence of any surface of absolute dis- continuity to divide the new parts from the original, so that the position given to the dividing surface is to a certain extent arbitrary. Even if the surface separating the masses were determined, the energy to be attributed to the masses separated would be partly arl)itrary, since a part of the total energy depends upon the mutual action of the two masses. We ought perhaps to consider the case the same in regard to the entropy, although the entropy of a system never depends upon the mutual relations of parts at sensible dis- tances from one another. Now the condition (52) will be valid if the quantities Df, 7>//, l>f, I)m^, Dm.-, . . . I))u„ are so defined that none of the assmuptious which have been made, tacitly or otherwise, relating to the formation of these new parts, shall be violated. These assumptions are the following: — that the relation between the varia- tions of the energy, entropy, volume, etc., of any of the original parts is not aifected by the vicinity of the new parts; and that the energy, entropy, volume, etc., of the system in its varied state are correctly represented by the sums of the energies, entropies, volumes, etc., of the various parts (original and new), so far at least as any of these quantities are determined or aftected by the formation of the new parts. We will suppose Z>f, Dij, iJv, Dm ^, Dni^ . . . Dm,, to be so defined that these conditions shall not be violated. This may be done in various ways. We may suppose that the jjosition of the surfaces separating the new and the original parts has been fixed in any suitable way. Tiiis Avill detej-mine the space and the matter belonging to the parts separated. If this does not determine the division of the entropy, we may suppose this determined in any suit- able arbitrary way. Thus we may suppose the total energy in and about any ne\v part to be so distributed that equation (12) as applied to the original parts shall not be violated by the formation of the new parts. Or, it may seem more simple to suppose that the imaginary surface which divides any new part from the original is so placed as to include all the matter which is affected by the vicinity of the new formation, so that the part or parts which we regard as original may be left homogeneous in the strictest sense, including uniform dentilties of eneryij and entropy., up to the very J. W. (Tihhs—I'JqKMlhriviii of Ueterof/eneoKS Substances. 131 bounding surface. The homogeneity of the new parts is of no con- sequence, as we have made no assumption in that respect. It may l)e doubtful whether we can consider the new parts, as thus hounded, to be infinitely small even in tlieir earliest stages of development. But if they are not infinitely small, the only way in which this can aftect the validity of our formuhe will be that in virtue of the equations of condition, i. e., in virtue of the evident necessities of the case, finite variations of the energy, entropy, volume, etc., of the original parts will be caused, to which it might seem that equation (12) would not apply. But if the nature and state of the mass be not varied, equa- tion (12) will hold true of finite dift'erences. (This appears at once, if we integrate the equation under the above limitation.) Hence, the equation will hold true for finite diiferences, provided that the nature and state of the mass be infinitely little varied. For the dif- ferences may be considered as made up of two parts, of which the first are for a constant nature and state of the mass, and the second are infinitely small. We may therefore regard the new parts to be bounded as supposed without prejudice to the validity of any of our results. The condition (52) understood in either of these ways (or in others which will suggest themselves to the reader) will have a per- fectly definite meaning, and will be valid as the necessary and sufii- cient condition of equilibi-ium in regard to the formation of new parts, when the conditions of equilibrium in regard to tlie original parts, (50), (51), and (43), are satisfied. In regard tf) the condition (53), it may be shown that with (50), (51), and (43) it is always suflicient for equilibrium. To prove this, it is only necessary to show that when (50), (51), and (43) are satis- fied, and (52) is not, (53) will also not be satisfied. We will first observe that an expression of the form _ e+ Tij- Pv^ J/, m^ + J/, "^2 • • • + -K i'^n (54) denotes the work olnainable V)y the formation (by a reversible pro- cess) of a body of which f, ;/, v, m^, in.^, . . . m„ are the energy, entropy, volume, and the quantities of the components, within a medium having the pressure P, the temperature 7] and tlie potentials 31 , J/2, . . . M„. (The medium is supposed so large that its prop- erties are not sensibly altered in any part by the formation of the body.) For f is the energy of the body formed, and the remaining terms represent (as may be seen by applying equation (12) to the medium) the decrease of the energy of the medium, if, after the 1:^2 J. TF. Gibbs—Eqailibviiim, of Heterogeneoiis Svbstances. formation of the body, the joint entropy of the medium and the hody, their joint volumes and joint quantities of matter, were the same as the entropy, etc., of the medium before the formation of the body. Tliis consideration may convince us that for any given finite values of v and of T, P, 31^ , etc. this expression cannot be infinite when f, //, m,, etc. are determined by any real body, whether homo- geneous or not, (but of the given volume), even when T, P, 3/j, etc. do not represent the values of the temperature, pressure, and poten- tials of any real substance. (If the substances *S',, /Sg, . . . S„ are all actual components of any homogeneous part of the system of which the equilibrium is discussed, that part will aiford an example of a body having the temperature, pressure, and potentials of the medium supposed.) Now by integrating equation (12) on the supposition that the nature and state of the mass considered i-emain unchanged, we obtain the equation which will hold true of any homogeneous mass whatever. Therefore for any one of the original parts, by (50) and (51), f - T)]-\-Pv-M^ m J - J/2 »«2 • • • — ^^n ^''„ = 0. (56) If the condition (52) is not satisfied in regard to all possible new parts, let JVhe a new part occurring in an original part O, for which the condition is not satisfied. It is evident that the value of the expression s—Ti] + Pv - M^ m^ — 31^ m^ . . . —3f„m„ (57) applied to a mass like 0 including some very small masses like JV, will be negative, and will decrease if the number of these masses like JV is increased, until there remains within the whole mass no portion of any sensible size without these masses like iV, which, it will be remembered, have no sensible size. But it cannot decrease without limit, as the value of (54) cannot become infinite. Now we need not inquire whether the least value of (57) (for constant values of T, P, M^, J/g* • • • -^^") would be obtained by excluding entirely the mass like 0, and filling the whole space considered with masses like iV, or whether a certain mixture would give a smaller value, — it is certain that the least possible value of (57) per unit of volume, and that a negative value, will be realized by a mass having a certain homogeneity. If the new part iVfor which the condition (52) is not satisfied occurs between two diflferent original parts 0' and 6>", the aigument need not be essentially varied. We may consider the J. W. Gihbs — Eqxiilihrium of Heterogeneous Substances. 133 value of (57). for u body consisting of masses like O' and 0" sepa- rated by a lamina i\^. This value may be decreased by^increasing the extent of this lamina, which may be done within a given volume by giving it a convoluted form ; and it will be evident, as before, that the least possible value of (57) will be for a homogeneous mass, and that the value will be negative. And such a mass will be not merely an ideal combination, but a body capable of existing, for as the expression (57) has for this mass in the state considered its least pos- sible value per unit of volume, the energy of the mass included in a unit of volume is the least possible for the same matter with the same entropy and volume, — hence, if confined in a non-conducting vessel, it will be in a state of not unstable equilibrium. Therefore when (50), (51), and (43) are satisfied, if the condition (52) is not sat- isfied in regard to all possible new parts, there will be some homo- geneous body which can be formed out of the substances aS'^, ^Sg, . . . S„ which will not satisfy condition (53). Therefore, if the initially existing masses satisfy the conditions (50), (51), and (43), and condition (53) is satisfied by every homoge- neous body which can be formed out of the given matter, there will be equilibrium. On the other hand, (53) is not a necessary condition of equilibrium. For we may easily conceive that the condition (52) shall hold true (for any very small formations within or between any of the given masses), while the condition (53) is not satisfied (for all large masses formed of the given matter), and experience shows that this is very often the case. Supersaturated solutions, superheated water, etc., are familiar examples. Such an equilibrium will, however, be practi- cally unstalde. By this is meant that, although, strictly speaking, an infinitely small disturbance or change may not be suflicient to destroy the equilibrium, yet a very small change in the initial state, perhaps a circumstance which entirely escapes our powers of percep- tion, will be sufficient to do so. The presence of a small portion of the substance for which tlie condition (53) does not hold true, is suffi- cient to produce this result, when this substance forms a variable component of the original homogeneous masses. In other cases, when, if the new substances are formed at all, different kinds must be formed simultaneously, the initial presence of the different kinds, and that in immediate proximity, may be necessary. It will be observed, that from (56) and (53) we can at once obtain (50) and (51), viz., by applying (53) to bodies differing infinitely little from the various homogeneous ])arts of the given mass. There- 134 ./ TK fribbs — Equilibrium of Heterogeneous Substances. fore, the condition (56) (relating to the various homogeneous parts of the given mass) and (53) (relating to any bodies which can be formed of the given matter) with (43) are always sufficient for equi- librium, and always necessary for an equilibrium which shall be practically stable. And, if we choose, we may get rid of limitation in regard to equations (43). For, if we compare these equations with (38), it is easy to see that it is always immaterial, in applying the tests (56) and (53) to any body, how we consider it to be com- posed. Hence, in applying these tests, we may consider all bodies to be composed of the ultimate components of the given mass. Then the terms in (56) and (53) which relate to other components than these will vanish, and we need not regard the equations (43). Such of the constants M ^, M.^ . . . 3I„ as relate to the ultimate compo ponents, may be regarded, like T and P, as unknown quantities sub- ject only to the conditions (56) and (53). These two conditions, which are sufficient for equilibrium and necessary for a practically stable equilibrium, may be united in one, viz., (if we choose the ultimate components of the given mass for the component substances to which Wj, w-g, . . . m^ relate) that it shall be possible to give such values to the constants T, P, J/j, J/2? . . . M^ in the expi'ession (o*?) that the value of the expression for each of the homogeneous parts of the mass in question shall be as small as for any body whatever made of the same components. Effect of Solidity of any Part of the given Mass. If any of the homogeneous masses of which the equilibrium is in question are solid, it will evidently be proper to treat the proportion of their components as invariable in the application of the criterion of equilibrium, even in the case of compounds of variable proportions., i. e., even when bodies can exist which are compounded in pro- portions infinitesimally varied from those of the solids considered. (Those solids which are capable of absorbing fluids form of course an exception, so far as their fluid components are concerned.) It is true that a solid may be increased by the formation of new solid matter on the surface where it meets a fluid, which is not homogeneous with the previously existing solid, but such a deposit will properly be treated as a distinct part of the system, (viz., as one of the parts which we have called new). Yet it is worthy of notice that if a homo- geneous solid which is a compound of variable proportions is in contact and equilibrium with a fluid, and the actual components of the solid (considered as of variable composition) are also actual com- J. W. Gihhs—Eqailibrli(m of Heterogeneous Substances. 135 poiients of the fluid, and tlie condition (53) is satisfied in regard to all bodies which can l)e formed out of the actual components of the fluid, (which will ahvaj-s be the case unless the fluid is practically unstable,) all the conditions will hold true of the solid, which would be necessary for equilibrium if it were fluid. This follows directly from the principles stated on the preceding pages. For in this case the value of (57) will be zero as determined either for the solid or for the fluid considered with reference to their ultimate components, and will not be negative for any body Avhatever which can be formed of these components; and these conditions are sufficient for equilibrium independently of the solidity of one of the masses. Yet the point is perhaps of sufficient importance to demand a more detailed consideration. Let xS„ . . . >% be the actual components of the solid, and aS'^, . . . S,, its possible components (which occur as actual components in the fluid); then, considering the proportion of the components of the solid as variable, we shall have for this body by equation (12) cW = t d)j - ^y civ' -f- //,/ dm J . . . H- //; dm.J + pi/dm^' . . . i-jutdn^. (58) By this equation the potentials j.ij . . . /u^.' are perfectly defined. But the difierentials dm„' . . . dmi.', considered as independent, evi- dently express variations w^hich are not possible in the sense required in the criterion of equilibrium. We might, however, introduce them into the genei-al condition of equilibrium, if we should express the dependence between them by the j^roper equations of condition. But it will be more in accordance with our method hitherto, if we consider the solid to have only a single independently variable com- ponent S^, of Avhich the nature is represented by the solid itself. We may then write 6e'=t' dif — p' dv' -f- jjj 6niJ. (59) In regard to the relation of the potential /^/ to the potentials occur- ring in equation (58) it will be observed, that as we have by integra- tion of (58) and (59) a' =: t' if - p' v' -\- /.(„' mj . . . + pij nij, (60) and e' = t' ?/ — p' v' + /jJ mj ; (61) therefore /.tj jt/J = /.tj mj . . . -\-f.i,'m,'. (62) Now, if the fluid has besides S^, . . . S,, and *S/, . . . S^. the actual components S/ . . . /S„, we may write for the fluid 130 J W. Gibbs — Eqailibriain of Heterogeneous Substances. + 11," Sm," . . . + /.It" 6m,r-{-pi/' dm/' . . . + //„" f^?«„", (63) and as by suppusition nij ®^ = mj ©„...+ »'*</' ®.v (6*) equations (43), (oO), and (51) will give in this ease on elimination of the constants T, P, etc., t'=:t", p'=p", (65) and mj M.' = '".,' I-'.." ■ • • +w^; //,/'• (66) Equations (65) and (66) may be regarded as expressing the condi- tions of equilibrium between the solid and the fluid. The last con- dition may also, in virtue of (62), be expressed by the equation w,,'//,,' . . . -j-n,; /.i; = mj /j„" . . . +';/*,>;'. (67) But if condition (53) holds true of all bodies which can be formed of «S'„ . . . S^, S,„ . . . iSi; S, . . . /8„, we may write for all such bodies £ — t" ?/-\-p" V — //„" m„ ... — //,/' m„ — //;," nh . . . — /V'w'i- — l-h' nil . . . M„"m„^ 0. (68) (In applying this formula to various bodies, it is to be observed that only the values of the unaccented letters are' to be determined by the different bodies to which it is applied, the values of the accented letters being already determined by the given fluid.) Now, by (60), (65), and (67), the value of the first member of this condition is zero when applied to the solid in its given state. As the condition must hold true of a body differing infinitesimally from the solid, we shall have dt' — t" dif -\-p" di^' — l^i„" dnij . . . ^" dnij — f.i,," dm,! ... - /V'fW= 0, (69) or, by equations (58) and (65), {l-i,,' — l^a') dm,; . . . -[_(//,/-;/;') c?;/,; + {Ih'-^u") dm,; ... 4- (/V-yWi") dm,'^ 0. (70) Therefore, as these differentials are all independent, ^,; = Ma", ■ ■ • mJ = mJ\ M>'= /■</'', ' • • Mh'^ /'x"; (71) which with (65) are evidently the same conditions which we would have obtained if we had neglected the fact of the solidity of one of the masses. J. W. Gibhs — Equilibrium of Heteroyentous /Substances. 1)37 We have supposed the solid to be homogeneous. But it is evident that in any case the above conditions must hold for every separate point where the solid meets the fluid. Hence, the temperature and pressure and the potentials for all the actual components of the solid must have a constant value in the solid at the surface where it meets the fluid. Now, these quantities are determined by the nature and state of the solid, and exceed in number the independent variations of which its nature and state ai'e capable. Hence, if we reject as improbable the supposition that the nature or state of a body can vary Avithout affecting the value of any of these quantities, we may conclude that a solid which varies (continuously) in nature or state at its surface cannot be in equilibrium with a stable fluid which con- tains, as independently variable components, the variable components of the solid. (There may be, however, in equilibrium with the same stable fluid, a finite number of different solid bodies, composed of the variable components of the fluid, and having their nature and state completely determined by the fluid.)* Effect of Additional Equations of Condition. As the equations of condition, of which we have made use, are such as always apply to matter enclosed in a rigid, impermeable, and non-conducting envelop, the particular conditions of equilibrium which we have found will always be sufficient tor equilibrium. But the number of conditions necessary for equilibrium, will be dimin- ished, in a case otherwise the same, as the number of equations of condition is increased. Yet the problem of equilibrium which has been treated will sufficiently indicate the method to be pursued in all cases and the general nature of the results. It will be observed that the position of the various homogeneous parts of the given mass, which is otherwise immaterial, may deter- mine the existence of certain equations of condition. Thus, when difterent parts of the system in which a certain substance is a vari- able component are entirely separated from one another by parts of which this substance is not a component, the quantity of this sub- stance will be invariable for each of the parts of the system which are thus separated, which will be easily expressed by equations of condi- tion. Other equations of condition may arise from the passive forces .(or resistances to change) inherent in the given masses. In the prob- * The solid has been considered as subject only to isotropic stresses. The effect of other stresses will be considered hereafter. Trans. Conn. Acad., Vol. III. 18 November, 1875. 138 X W. Gibbs—Equilihriuin of Heterogeneous Substcmces. lem which we are next to consider there are eqnations of condition due to a cause of a different nature. Eff'ect of a Diaphragm {EqniUbrmm of Osmotic Forces). If the given mass, enclosed as before, is divided into two parts, each of which is homogeneous and fluid, by a diaphragm which is capable of supporting an excess of pressure on either side, and is per- meable to some of the components and impermeable to others, we shall have the equations of condition 6,f-\-6v"=% (72) (W=iO, 6v"=0, (73) and for the components which cannot pass the diaphragm 6mJ=0, dmj'=0, Sm,,' = 0, Sm,," z=0, etc., (74) and for those which can dm,,' + d)j/,"= 0, Sm/ -f Stn/' = 0, etc. (75) With these equations of condition, the general condition of equilib- rium (see (15)) will give the following particular conditions: t' = t", (76) and for the components which can pass the diaphragm, if actual com- ponents of both masses, /'//=/'/', Mt'=^h", etc., (77) but not 2^' = p" ■> nor iA,lz=if.i^\ i.(f; = ii,'\ etc. Again, if the diaphragm is pei'meable to the components in certain proportions only, or in proportions not entirely determined yet sub- ject to certain conditions, these conditions may be expressed by equations of condition, which will be linear equations between 6m^\ Sm^'t etc., and if these be known the deduction of the i^articular con- ditions of equilibrium will present no difficulties. We will however observe that if the components aS',, S2, etc. (being actual components on each side) can pass the diaphragm simultaneously in the propor- tions a J, a^, etc. (without other resistances than such as vanish with the velocity of the current), values proportional to a^, a^, etc. are possible for dni^\ Sm^', etc. in the general condition of equilibrium, 6m ^", Sm^"^ etc. having the same values taken negatively, so that we shall have for one particular condition of equilibrium ^1 /'/+ "2 '"2' + ^^^- — '-^1 " 1" + ^h Ih" -^ etc. (78) There will evidently be as many independent equations of this form J. W. Gibbs — Equilibrium of Heterogeneous Substances. 139 as there are independent combinations of the elements which can pass the diaj^hragra. These conditions of equilibrium do not of course depend in any way upon the supposition that the volume of each fluid mass is kept constant, if the diaphragm is in any case supposed immovable. In fact, we may easily obtain the same conditions of equilibrium, if we suppose the volumes variable. In this case, as the equilibrium must be preserved by forces acting upon the external surfaces of the fluids, the variation of the energy of the sources of these forces must appear in the general condition of equilibrium, which will be 6t'-\-6e" -^P' dv'-ifP" SV'^O, (79) JP and P" denoting the external forces per unit of area. (Compare (14).) From this condition we may evidently derive the same internal conditions of equilibrium as before, and in addition the external conditions p' — P\ p"z=P". (80) In the preceding paragraphs it is assumed that the permeability of the diajjhragm is perfect, and its impermeability absolute, i. e., that it offers no resistance to the passage of the components of the fluids in certain proportions, except such as vanishes with the velocity, and that in other proportions the components cannot pass at all. How far these conditions are satisfied in any particular case is of course to be determined by experiment. If the diaphragm is permeable to all the n components without restriction, the temperature and the potentials for all the components must be the same on both sides. Now, as one may easily convince himself, amass having n components is capable of only /i + 1 inde- pendent variations in nature and state. Hence, if the fluid on one side of the diaphragm remains without change, that on the other side cannot (in general) vary in nature or state. Yet the pressure will not necessarily be the same on both sides. For, although the pres- sure is a function of the temperature and the n potentials, it may be a many-valued function (or any one of several functions) of these variables. But when the pi-essures are different on the two sides, the fluid which has the less pressure will be practically unstable, in the sense in which the term has been used on page 133. For £"_?;" 7/' +/>"?/'—/.</'»*," — /1 2" ??^2" . . . — //„"/>/„" = 0, (SI) as appears from equation (12) if integrated on the supposition that the nature and state of the mass remain unchanged. Therefore, if p<}j" while t'-t'\ ;t,'= ;:,", etc. 140 J. W. Gibbs — Equilibrmm of Heteroge)ieous ^Substances. f," _t' //' -\-p'v"- /.i^'m," - J^to'ms" . . . - /-'n m„" <^0. (82) This relation indicates the instability of tlie fluid to which the single accents refer. (See page 133.) But independently of any assumption in regard to the permeability of the diaphragm, the following relation will hold true in any case in which eacli of the two fluid masses may be regarded as unifonn throughout in nature and state. Let the character d be used with the variables which express the nature, state, and quantity of the fluids to denote the increments of the values of these quantities actu- ally occurring in a time either flnite or infinitesimal. Then, as the heat received by the two masses cannot exceed t'T>}/ -\-t" v>if', and as the increase of their energy is equal to the difference of the heat they receive and the work they do, Di' + T>b" -St' litf + «"d//' — />'du'— p"iyv", (83) i.e., by (12), yu,'Dm,'+/(i"Dm/' + //2'n?;4' + /<2"Dm2" + etc. ^0, (84) or (///' — ///) r.m/'+ (/^2"-/^2') ^>m,"+ etc. ^0. (85) It is evident that the sign = liolds true only in the limiting case in which no motion takes place. DEFINITION AND PROPERTIES OF FUNDAMENTAL EQUATIONS. The solution of the problems of equilibrium which we have been considering has been made to depend upon the equations which express the relations between the energy, entropy, volume, and the quantities of the various components, for homogeneous combinations of the substances which are found in the given mass. The nature of such equations must be determined by experiment. As, however, it is only differences of energy and of entropy that can be measured, or indeed, that have a physical meaning, the values of these quantities are so far arbitrary, that we may choose independently for each simple substance the state in which its energy and its entropy are both zero. The values of the energy and .the entropy of any com- pound body in any particular state will then be fixed. Its energy will be the sum of the work and heat expended in bringing its com- ponents from the states in which their energies and their entropies are zero into combination and to the state in question ; and its entropy is the value of the integral / — for any reversible process J. W. Gibbs — Equilibrium of Hetei'ogeiieoiis Substances. 141 by which that change is effected [dQ denoting an element of the heat communicated to the matter thns treated, and t the temperature of the matter receiving it). In the determination botli of the energy and of the entropy, it is understood that at the close of the process, all bodies whicli have been used, other than those to which the deter- minations relate, have been restored to their original state, with the exception of the sources of the work and heat expended, which must be used only as such sources. We know, however, a priori, that if the quantity of any homoge- neous mass containing it. independently variable components varies and not its nature or state, the quantities f, ?/, v, »i,, m^, . . . ni„ will all vary in the same proportion ; therefore it is sufficient if we learn from experiment the relation between all but any one of these quan- tities for a given constant value of that one. Or, we may consider that we have to learn from experiment the relation subsisting between the n i- 2 ratios of the n -{- 3 quantities f, //, v, m^, ra^, . . . m„. To fix our ideas we may take for these ratios , -, — ?, — -. etc., that is, the separate densities of the components, and the ratios £ If - and -, which may be called the densities of energy and entropy. But when there is but one comj^onent, it may be more convenient to choose — , — , — as the three variables. In any case, it is only a func- m ni, ni j 7 ., tion of w. -f- 1 independent variables, of which the form is to be deter- mined by experiment. Now if £ is a known function of ;/, w, m^, m^, . . . m^, as by equa- tion (12) de-=.td)] - p dv + // , dm j -|- /ig ^^2 • • • + /v„ dm„, (86) t,p,' 1^1, ^2') • • • A'n ^'"^ functions of the same variables, which may be derived from the original function by differentiation, and may therefore be considered as known functions. This will make n -\- S independent known relations between the 2n + 5 variables, e, /;, v m^, 7712, • • • "^n» t,P, /-^i-, 1^2, ■ ■ ■ /'n- These are all that exist, for of these variables, n + 2 are evidently independent. Now upon these relations depend a very large class of the properties of the compound considered, —we may say in general, all its thermal, mechanical, and chemical properties, so far as active tendencies are concerned, in cases in which the form of the mass does not require consideration. A single equation from which all these relations may 142 J. W. Gihbs —Equilibrluiii. of Heterogeneous Substances. be deduced we will call a fundamental equation for the substance in question. We shall hereafter consider a more general form of the fun- damental equation for solids, in which the pressure at any point is not supposed to be the same in all directions. But for masses subject only to isotropic stresses an equation between f, //, w, m^,m^, . . . m„ is a fundamental equation. There are other equations which possess this same property.* Let '/'=f-^'A (87) then by differentiation and comparison with (86) we obtain d ij' =z — i/dt — pdv -f- /^j dni^ -\- m^ dm^ . . . + /.i^dm^. (88) If, then, y- is known as a function of t, v, m^, m.^, . . . m„, we can find If, p, J-i 1, /'■>, • ■ • A'n i" terms of the same variables. If we then substitute for //' in our original equation its value taken from eq. (87), we shall have again 7i -\- 3 independent relations between the same 2n + 5 variables as before. Let X=£+pv, (89) then by (86), dx — tdi] + V dp 4-/^1 dm^ + //g dm^ ... -|- //„ drn^. (90) If, then, X be known as a function of }i,p, m^, m.^, . , . rn„, we can find t, V, yUj, /<2» • • • /^n i" terms of the same variables. By elimi- nating J, we may obtain again n + 3 independent relations between the same 2?/ + 5 variables as at first. Let ^ = e - ttf +pv, (91) then, by (86) di^=:. — ffdt + V dp + ;<j dm^ + 1.(2 dm „ . . . + ^^dm^. (92) If, then, ^ is known as a function of ^, /?, mj, mg, . . . ;;?„, we can * M. Massieu (Comptes Rendus, T. Ixix, 1869, p. 858 and p. 1057) has shown how all the properties of a fluid " which are considered in thermodynamics" may be deduced from a single function, which he calls a characteristic function of the fluid considered. In the papers cited, he introduces two different functions of this kind ; viz., a function of the temperature and volume, which he denotes by 1/), the value of — t + tn ~ f which in our notation would be 7 or — r— ; and a function of the temperature and pressure, which he denotes by V^', the value of which in our notation would be — e + tr/ —pv — C 1 or -7-. In both cases he considers a constant quantity (one kilogram) of the fluid, which is regarded as invariable in composition. J. W. Gibbs — JEquilibrlum of Heterogeneous Substances. 143 find If, V, yUj, /.i.j,, • • • Mn i^i terms of the same variables. By elimi- nating C, we may obtain again n -{- S independent relations between the same 2n + 5 variables as at first. If we integrate (86), supposing the quantity of the compound sub- stance considered to vary from zero to any finite value, its nature and state remaining unchanged, we obtain s=ztff — pv + /^ 1 in J + //^ »« 3 . . . + //„ ?n„, (93 ) and by (87), (89), (91) Tlie last three equations may also be obtained directly by integrating (88), (90), and (92). If we differentiate (93) in the most general manner, and compare the result with (86), we obtain — V dp -\- tjdt + m^ dfi^ -\- in^ dji^ . . . + )n„dii„-=. 0, (97) or dp=i- dt H i <?/<! H df.i^ . . . H df.1^ = 0. (98) Hence, there is a relation between the n + 2 quantities t, p, jli^, fi.^, . . . yt/„, which, if known, will enable us. to find in terms of these quan- tities all the ratios of the n + 2 quantities //, v, m^, m^ . . . m„. With (93), this will make n + S independent relations between the same 2n + 5 variables as at first. Any equation, therefore, between the quantities + /v„ m„. (94) + Mn w„, (95) + //„m„. (96) f, V, V, m„ ■/7<3, . . . rn„, (99) or '/', f, V, mj. m^, . . • m„, (100) or A', V, Ih rn^, ^2, . , . . m„, (101) or ^, t, P. mi, "*2, • ■ ' m„, (102) or t, P, /<i, l-lo, . . ■ l-ln, (103) is a fundamental equation, and any such is entirely equivalent to any other.* For any homogeneous mass whatever, considered (in gen- eral) as variable in composition, in quantity, and in thermodynamic state, and having n independently variable components, to which * The distinction between equations which are, and which are not, fundamental, in the sense in whicli the word is here used, may be illustrated by comparing an equation 144 J. W. Glhbs —Equilibriuin of Heterogeneous Substances. the subscript nuraertils refer, (but not excluding tlie case in which // z= 1 and the composition of tlie body is invariable,) there is a rela- tion between the quantities enumerated in any one of the above sets, from which, if known, with the aid only of general principles and relations, we may deduce all the relations subsisting for such a mass between the quantities e, i/-, x, I', '/, v, rjt ^, m.^, . . , ///„, t, p, /.ij, ju.^, . . . //„. It will be observed that, besides the equations which define i/:, x, and 'C, there is one finite equation, (93), which subsists between these qiiantities independently of the form of the fundamental equation. Other sets of quantities might of course be added which possess the same property. The sets (100), (101), (102) are mentioned on account of the important properties of the quantities i/-, j, 'Q, and because the equations (88), (90), (92), like (86), ufiTord convenient definitions of the potentials, viz., ;,,=(*) ={'PL) =(m =(^) (104) etc., where the subscript letters denote the quantities which remain constant in the differentiation, m being written for brevity for all the letters m^, mg, . . . in„ except the one occurring in the denominator. It will be observed that the quantities in (103) are all independent of the quantity of the mass considered, and are those which must, in general, have the same value in contiguous masses in equilibrium. 0)1 the quantities i/\ j, t. The quantity //' has been defined for any homogeneous mass by the equation if' -€ ~ tt]. (105) between e, J], V, m,, m,, . . . Win with one between c, t,v, m,, mo, . ■ . in„. As, by (86), de ) the second equation may evidently be derived from the first. But the first equation cannot be derived from the second; for an equation between ^' \dn) - ^. "in w?) • • • w„ Kd?! is equivalent to one between drj de f, V, m,, m.^, . . . m„. which is evidently not sufficient to determine the value of ?/ in terms of the other variables. J. W. Gihbs — Equilibrium of Heterogeneous Substances. 145 We may extend this definition to any material system whatever which has a nniform temperature throughout. If we compare two states of the system of the same temperature, we have f-f' = f'- 6" -?;(;/ -,/'). (106) If we suppose the system brought from the first to the second of these states without change of temperature and by a reversible pro- cess in which W is the work done and Q the heat received by the system, then £'-f":rrTF- Q, (107) and t{if' ^ i/)=Q. (108) Hence //''-//'" = IF; (109) and for an infinitely small reversible change in the state of the system, in which the temperature remains constant, we may write ~dip = dW. (110) Therefore, — //' is the force function of the system for constant temperature, just as — £ is the force function for constant entropy. That is, if we consider if: as a function of the temperatm-e and the variables which express the distribution of the matter in space, for every different value of the temperature — ?/' is the different force function required by the system if maintained at that special temperature. From this we may conclude that when a system has a uniform temperature throughout, the additional conditions which are necessary and sufficient for eqiiilibrium may be expressed by (d^')<^0-* (111) * This general condition of equilibrium might be used instead of (2) in such prob- lems of equilibrium as we have considered and others which we shall consider here- after with evident advantage in respect to the brevity of the formulas, as the limitation expressed by the subscript i in (111) applies to every part of the system taken sepa- rately, and diminishes by one the number of independent variations in the state of these parts which we have to consider. The more cumbersome course adopted in this paper has been chosen, among other reasons, for the sake of deducing all the particular conditions of equilibrium from one general condition, and of having the quantities mentioned in this general condition such as are most generally used and most simply defined ; and because in the longer formulae as given, the reader will easily see in each case the form which they would take if we should adopt (111) as the general condi- tion of equilibrium, which would be in effect to take the thermal condition of equilibrium for granted, and to seek only the remaining conditions. For example, in the problem treated on pages 116 fE., we would obtain from (111) by (88) a condition precisely like (15), except that the terms 16?]', tSrj"^ etc. would be wanting. Trans. Conn. Acad., Vol. III. 19 January, 1876. 146 ./. W. Glbbs — Equilibriion of IIete)'0(jeneous SubsUinces. When it is not possible to bring the system from one to the other of the states to which //'' and '/'" relate by a reversible process without altering the temperature, it will be observed that it is not necessary for the validity of (107)-(109) that the temperature of the system should remain constant during the reversible process to which TTand Q relate, provided that the only source of heat or cold used has the same temperature as the system in its initial or final state. Any external bodies may be used in the process in any Avay not affect- ing the condition of reversibility, if restored to their original con- dition at the close of the process ; nor does the limitation in regard to the use of heat apply to such heat as may be restored to the source from which it has been taken. It may be interesting to show directly the equivalence of the condi- tions (111) and (2) when applied to a system of which the temperature in the given state is uniform throughout. If there are any variations in the state of such a system which do not satisfy (2), then for these variations 6e<Q and 6}] = Q. If the temperature of the system in its varied state is not uniform, we may evidently increase its entropy without altering its energy by supposing heat to pass from the warmer to the cooler parts. And the state having the greatest entropy for the energy f -|- (Je will necessarily be a state of uniform temperature. For this state (regarded as a variation from the original state) dE<Q and 6i]>Q. Hence, as we may diminish both the energy and the entropy by cool- ino- the system, there must be a state of uniform temperature for which (regarded as a variation of the original state) rff < 0 and (J// = 0. From this we may conclude that for systems of initially uniform tem- perature condition (2) will not be altered if we limit the variations to such as do not disturb the uniformity of temperature. Confining our attention, then, to states of uniform temperature, we have by differentiation of (105) 6s - tdi}=dil^-\-})dt. (112) Now there are evidently changes in the system (produced by heating or cooling) for which de - t (h/ = 0 and therefore Si/^ -[-7jdt=:0, (113) J. W. Gibbs — Equilibrium of Heterogeneous Substances. 147 neither S/; nor dt having the value zero. This consideration is suffi- cient to show that the condition (2) is equivalent to de — tdf/^0. (114) and that the condition (111) is equivalent to Sif^-^}/6t^0 . (115) and by (112) the two last conditions are equivalent. In such cases as we have considered on pages 115-137, in which the form and position of the masses of which the system is composed is immaterial, uniformity of temperature and pressure are always necessary for equilibrium, and the remaining conditions, when these are satisfied, may be conveniently expressed by means of the func- tion ?, which has been defined for a homogeneous mass on page 142, and which we will here define for any mass of uniform temperature and pressure by the same equation t,^£ — ttj-\-pv. (Ii6) For such a mass, the condition of (internal) equilibrium is m,,^o. (117) That this condition is equivalent to (2) will easily appear from con- siderations like those used in respect to (111). Hence, it is necessary for the equilibrium of two contiguous masses identical in composition that the values of C as determined for equal quantities of the two masses should be equal. Or, when one of three contiguous masses can be formed out of the other two, it is necessary for equilibrium that the value of C for any quantity of the first mass should be equal to the sum of the values of t. for such quantities of the second and third masses as together contain the same matter. Thus, for the equilibrium of a solution composed of a parts of water and b parts of a salt which is in contact with vapor of water and crystals of the salt, it is necessary that the value of t, for the quantity a-\-b oi the solution should be equal to the sum of the values of C for the quanti- ties a of the vapor and b of the salt. Similar propositions will hold true in more complicated cases. The reader will easily deduce these conditions from the particular conditions of equilibrium given on page 128. In like manner we may extend the definition of x to any mass or combination of masses in which the pressure is everywhere the same, using e for the energy and v for the volume of the whole and setting as before X=e-\-pv. (118) 148 J. W. Gibbs — Eqidlibrium of Heterogeneous Substances. If we denote by Q the heat received by the combined masses from external sources in any process in which the pressure is not varied, and distinguish the initial and final states of the system by accents we have /' - / = 6" - 6' +p {v" - v') = Q. (119) This function may therefore be called the heat function for constant pressure (just as the energy might be called the heat function for constant volume), the diminution of the function representing in all cases in which the pressure is not varied the heat given out by the system. In all cases of chemical action in which no heat is allowed to escape the value of j remains unchanged. POTENTIALS. In the definition of the potentials /i^, /Yg, etc., the energy of a homogeneous mass was considered as a function of its entropy, its volume, and the quantities of the various substances composing it. Then the potential for one of these substances was defined as the dif- ferential coefficient of the energy taken with respect to the variable expressing the quantity of that substance. Now, as the manner in which we consider the given mass as composed of various substances is in some degree arbitrary, so that the energy may be considered as a function of various different sets of variables expressing quantities of component substances, it might seem that the above definition does not fix the value of the potential of any substance in the given mass, until we have fixed the manner in which the mass is to be con- sidered as composed. For example, if we have a solution obtained by dissolving in water a certain salt containing water of crystalliza- tion, we may consider the liqviid as composed of nig weight-units of the hydrate and myy of water, or as composed of m, of the anhydrous salt and w„, of water. It will be observed that the vahies of m,, and m, are not the same, nor those of m„- and m,,,, and hence it might seem that the potential for water in the given liquid considered as composed of the hydrate and water, viz., (^\ \d»ijy/fi, V, ms would be different from the potential for water in the same liquid con- sidered as composed of anhydrous salt and water, viz., J. W. Gihbs — EquiUhrlum of Heterogeneous ISuhstances. 149 The value of the two expressions is, however, the same, for, although ?>?„• is not equal to w„,, we may of course suppose chuyy to he equal to clm^, and then the numerators in the two fractions will also be equal, as they each denote the increase of energy of the liquid, when the quantity (hn^y or drn^, of water is added without altering the entropy and volume of the liquid. Precisely the same considerations will apply to any other case. In fact, we may give a definition of a potential which shall not pre- suppose any choice of a particular set of substances as the components of the homogeneous mass considered. Definition. — If to any homogeneous mass we suppose an infinitesi- mal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass con- sidered. (For the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a substance, whether capable or not of existing by itself as a homoge- neous body.) In the above definition we may evidently substitute for entropy, volume, and energy, respectively, either temperature, volume, and the function ij- ; or entropy, pressure, and the function x ; or tempera- ture, pressure, and the function ;;. (Compare equation (104).) In the same homogeneous mass, therefore, we may distinguish the potentials for an indefinite number of substances, each of which has a perfectly determined value. Between the potentials for diiferent substances in the same homo- geneous mass the same equations will subsist as between the units of these siibstances. That is, if the substances, *S„, /S',„ etc., ^S^, Si, etc., are components of any given homogeneous mass, and are such that a 2„ + /^ g, + etc. = n ©, + ^ ©/+ etc., (120) ©a, S45 etc., 2i, S/, etc. denoting the units of the several substances, and «, /j, etc., «, A, etc. denoting numbers, then if /<„, ^,„ etc., /z^, /^„ etc. denote the potentials for these substances in the homogeneous mass, a i-ia + /^ yWi + etc. = H f-ik + A /^, + etc. (121) To show this, we will suppose the mass considered to be very large. Then, the first number of (121) denotes the increase of the energy of the mass produced by the addition of the matter represented by the first member of (120), and the second member of (121) denotes the 150 J. W. Gibbs — Equilibrmrn of Heterogeneous Substances. increase of energy of the same mass produced by the addition of the matter represented by the second member of (120), the entropy and volume of the mass remaining in each case unchanged. Therefore, as the two members of (120) represent the same matter in kind and quantity, the two members of (121) must be equal. But it must be understood that equation (120) is intended to denote equivalence of the substances represented in the mass con- sidered, and not merely chemical identity ; in other words, it is sup- posed that there are no passive resistances to change in the mass considered which prevent the substances represented by one member of (120) from passing into those represented by the other. For example, in respect to a mixture of vapor of water and free hydrogen and oxygen (at ordinary temperatures), we may not write but water is to be treated as an independent substance, and no neces- sary relation will subsist between the potential for water and the potentials for hydrogen and oxygen. The reader will observe that the relations expressed by equations (43) and (51) (which are essentially relations between the poten- tials for actual components in different parts of a mass in a state of equilibrium) are simply those which by (121) would necessary sub- sist between the same potentials in any homogeneous mass containing as variable components all the substances to which the potentials relate. In the case of a body of invariable composition, the potential for the single component is equal to the value of t, for one unit of the body, as appears from the equation 1;=: /.nn (122) to which (96) reduces in this case. Therefore, when n = \, the fun- damental equation between the quantities in the set (102) (see page 143) and that between the quantities in (103) may be derived either from the other by simple substitution. But, with this single excep- tion, an eqiiation between the quantities in one of the sets (99)-(103) cannot be derived from the equation between the quantities in another of these sets without differentiation. Also in the case of a body of variable composition, when all the quantities of the components except one vanish, the potential for that one will be equal to the value of t, for one unit of the body. We may make this occur for any given composition of the body by J. W. Gibbs — Equilibrium of Heterogeneous Substances. 15] choosing as one of the components the matter constituting the body itself, so that the value of ? for one unit of a body may always be considered as a potential. Hence the relations between the values of ? for contiguous masses given on page 1 47 may be regarded as rela- tions between potentials. The two following propositions afford definitions of a potential which may sometimes be convenient. The potential for any substance in any homogeneous mass is equal to the amount of mechanical work required to bring a unit of the substance by a reversible process from the state in which its energy and entropy are both zei'o into combination with the homogeneous mass, which at the close of the process must have its original volume, and which is supposed so large as not to be sensibly altered in any part. All other bodies used in the process must by its close be restored to their oi'iginal state, except those used to supply the work, which must be used only as the source of the work. For, in a reversible process, when the entropies of other bodies are not altered, the entropy of the substance and mass taken together will not be altered. But the original entropy of the substance is zero; therefore the entropy of the mass is not altered by the addition of the substance. Again, the work expended will be equal to the increment of the energy of the mass and substance taken together, and therefore equal, as the original energy of the substance is zero, to the increment of energy of the mass due to the addition of the substance, which by the definition on page 149 is equal to the potential in question. The potential for any substance in any homogeneous mass is equal to the work required to bring a unit of the substance by a reversible process from a state in which //' = 0 and the temperature is the same as that of the given mass into combination with this mass, which at the close of the process must have the same volume and temperature as at first, and which is supposed so large as not to be sensibly altered in any part. A source of heat or cold of the temperature of the given mass is allowed, with this exception, other bodies are to be used only on the same conditions as before. This may be shown by applying equation (109) to the mass and substance taken together. The last proposition enables us to see very easily, how the value of the potential is affected by the arbitrary constants involved in the definition of the energy and the entropy of each elementary sub- stance. For we may imagine the substance brought from the state in which tp=zQ and the temperature is the same as that of the given 152 J.W. Gibhs — Equilibrium of Heterogeneous Substances. mass, first to any specified state of the same temperature, and then into combination with the given mass. In the first part of the pro- cess the work expended is evidently represented by the value of y.' for the unit of the substance in the state specified. Let this be denoted by </'', and let /< denote the potential in question, and W the work expended in bringing a unit of the substance from the specified state into combination with the given mass as afoi-esaid ; then lx=ip'-^W. (123) Now as the state of the substance for which 6=0 and ?/ = 0 is arbitrary, we may simultaneously inci-ease the energies of the unit of the substance in all possible states by any constant C, and the entropies of the substance in all possible states by any constant K. The value of //•, or £ — t //, for any state would then be increased by C -^ t K, t denoting the temperature of the state. Applying this to if:' in (123) and observing that the last term in this equation is independent of the values of these constants, we see that the potential would be increased by the same quantity C — t K, t being the tem- perature of the mass in which the potential is to be determined. ON COEXISTENT PHASES OF MATTER. In considering the different homogeneous bodies which can be formed out of any set of component substances, it will be convenient to have a term which shall refer solely to the composition and ther- modynamic state of any such body without regard to its quantity or form. We may call such bodies as differ in composition or state dif- ferent phases of the matter considered, regarding all bodies which differ only in quantity and form as different examples of the same phase. Phases which can exist together, the dividing surfaces being plane, in an equilibrium which does not depend upon passive resist- ances to change, we shall call coexistent. If a homogeneous body has n independently variable components, the phase of the body is evidently capable of n. -|- 1 independent vari- ations. A system of r coexistent phases, each of which has the same n independently variable components is capable of « + 2 — r varia- tions of phase. For the temperature, the pressure, and the poten- tials for the actual components have the same values in the different phases, and the variations of these quantities are by (97) subject to as many conditions as there are different phases. Therefore, the num- ber of independent variations in the values of these quantities, i. e., the number of independent variations of phase of the system, will be n+2 -r. J, W. Gibhs — Equilibrmm of Heteroyeneous Substances. 158 Or, when the r bodies considered have not the same independently- variable components, if we still denote by n the number of indeperud- ently variable components of the r bodies taken as a whole, the number of independent variations of phase of which the system is capable will still be w+2 — r. In this case, it will be necessary to consider the potentials for more than 71 component substances. Let the number of these potentials be n-\-h. We shall have by (97), as before, r relations between the variations of the temperature, of the pressure, and of these n-^h potentials, and we shall also have by (43) and (51) h relations between these potentials, of the same form as the relations which subsist between the units of the different component substances. Hence, if r = w + 2, no variation in the phases (remaining coex- istent) is possible. It does not seem probable that r can ever exceed n + 2. An example of nz=.\ and rz=.Z is seen in the coexistent solid, liquid, and gaseous forms of any substance of invariable composition. It seems not improbable that in the case of sulphur and some other simple substances there is more than one triad of coexistent phases ; but it is entirely improbable that there are four coexistent phases of any simple substance. An example of /i = 2 and r-=.4: \s seen in a solution of a salt in water in contact with vapor of water and two different kinds of crystals of the salt. Concerning n -{- \ Coexistent Phases. We will now seek the differential equation which expresses the relation between the variations of the tem})erature and the pressure in a system of w -f 1 coexistent phases [n denoting, as before, the number of independently variable components in the system taken as a whole). In this case we have n + 1 equations of the general form of (97) (one for each of the coexistent phases), in which we may distinguish the quantities //, v, m^, Wg, etc. relating to the different phases by accents. But t and^ will each have the same value throughout, and the same is true of /Vj, /'g, etc., so far as each of these occurs in the different equations. If the total number of these potentials is n + h, there will be h independent relations between them, corresponding to the h independent relations between the units of the component sub- stances to which the potentials relate, by means of which we may eliminate the variations of h of the potentials from the equations of the form of (97) in which they occur. Trans. Conn. Acad., Vol. III. 20 January, 1876. 154 J. W. Gihbs — Equilibrium of Heterogeneous Substances. Let one of these equations be v' dp=z 7]' dt + mj djj>a + "Tin-i df.ii + etc., (124) and by the proposed elimination let it become v' dp= If dt + A^' dj-i^ + A^ df.i2 . . . + A„' d/.j„. (125) It will be observed that //„, for example, in (124) denotes the poten- tial in the mass considered for a substance *S'„ which may oi- may not be identical with any of the substances S^, S2, etc. to which the potentials in (125) relate. Now as the equations between the poten- tials by means of which the elimination is performed are similar to those which subsist between the units of the corresponding sub- stances, (compare equations (38), (43), and (51),) if we denote these units by (Sa, ©4, etc., ©i, ©g, etc., we must also have m„'(S„ + m;®,, + etc. = .4j'(5i-|-^2'®2 • • • +^J„'®„- (126) But the first member of this equation denotes (in kind and quantity) the matter in the body to which equations (1 24) and (125) relate. As the same must be true of the second memV)er, we may regard this same body as composed of the quantity A ^' of the substance aS,, with the quantity A^' of the substance 1S2, etc. We will therefore, in accord- ance with our general xisage, write m^' tn^', etc. for A^', A2', etc. in (125), which will then become v' dp = 7/ dt -f »2i' d/.i^ + m^ dyi^ • • • + "*«' d^^- (127) But we must remember that the components to which the m/, mg', etc. of this equation relate are not necessarily independently variable, as are the components to which the similar expressions in (9V) and (124) relate. The rest of the /i + 1 equations may be reduced to a similar form, viz., v" dp = 7f dt -\- m^" di-i^ ■^-m^'dj.i^ . ■ . +m„"dp„, (128) etc. By elimination of f?/< j, d/^i^, . . . dfi„ from these equations we obtain v m v" m v'" rn '"n n m.^ nis m: v" -/' ?«2 m„"' dp =. v'" m/" rn^ dt. (129) In this equation we may make v', v", etc. equal to unity. Then m,', mg', m/', etc. will denote the separate densities of the compo- nents in the different phases, and //', ?/', etc. the densities of entropy. J. W. Gibbs — Equilibrium of Heterogeneous Substances. 155 When n=L 1, {m" v' ~ m' v") dp = {m" if - m //") dt, (130) or, if we make ni' =. 1 and m" =. 1, we liave the usual formula dt v'-v" t{v"-v'y ^ ^ in which Q denotes the heat absorbed by a unit of the substance in passing from one state to the other without change of temperature or pressure. Co7icerning Cases in which the Number of Coexistent Phases is less than /i-J- 1. When M> 1, if the quantities of all the components /S'j, /Sg, . . . S„ are proportional in two coexistent phases, the two equations of the form of (127) and (128) relating to these phases will be sufficient for the elimination of the variations of all the potentials. In fact, the condition of the coexistence of the two phases together with the condition of the equality of the n — 1 ratios of «*/, m^ ^ , . . m„' with the n — \ ratios of m^\ ^'^-z" ■> • • • ''*"' ^^ sufficient to detei'mine /> as a function of t if the fundamental equation is known for each of the phases. The ditferential equation in this case may be expressed in the form of (130), m' and m" denoting either the quantities of any one of the components or the total quantities of matter in the bodies to which they relate. Equation (131) will also hold true in this case, if the total quantity of matter in each of the bodies is unity. But this case differs from the preceding in that the matter which absorbs the heat Q in passing from one stat j to another, and to which the other letters in the formula relate, alt-iough the same in quantity, is not in general the same in kind at different temperatures and pressures. Yet the case wall often occur that one of the phases is essentially invariable in composition, especially when it is a crystalline body, and in this case the matter to which the letters in (131) relate will not vary with the temperature and pressure. When n = 2, two coexistent phases are capable, when the temper- ature is constant, of a single variation in phase. But as (130) will hold true in this case when m/ : m^' : : m^" : m^", it follows that for constant temperature the pressure is in general a maximum or a min- imum when the composition of the two phases is identical. In like manner, the temperature of the two coexistent phases is in general a maximum or a minimum, for constant pi-essure, when the composition of the two phases is identical. Hence, the series of simultaneous values of t and p for which the composition of two coexistent phases 156 J. W. Gibhs — Equilibrium of Heterogeneous Substances. is identical separates those simultaneous values of t and p for which no coexistent phases are possible from those for which there are two pair of coexistent phases. This may be applied to a liquid having two independently variable compouents in connection with the vapor which it yields, or in connection with any solid which may be formed in it. When n = 3, we have for three coexistent phases three equations of the form of (127), from which we may obtain the following, V rn v" m v'" rn dp=i dt--\- m. tn. djJi^. (132) Now the value of the last of these determinants will be zero, when the composition of one of the three phases is such as can be produced by combining the other two. Hence, the pressure of three coexistent phases will in general be a maximum or minimum for constant tem- perature, and the temperature a maximum or minimvim for constant pressure, when the above condition in regard to the composition of the coexistent phases is satisfied. The series of simultaneous values of t and p for which the condition is satisfied separates those simul- taneous values of t and p for which three coexistent phases are not possible, from those for which there are two triads of coexistent phases. These propositions may be extended to higher values of ;i, and illustrated by the boiling temperatures and pressures of saturated solutions of ?^ — 2 different solids in solvents having two independ- ently variable components. INTERNAL STABILITY OF HOM()(iENEOUS FLUIDS AS INDICATED BY FUNDAMENTAL EQUATIONS. We will now consider the stability of a fluid enclosed in a rigid envelop which is non-conducting to heat and impermeable to all the components of the fluid. The fluid is supposed initially homogeneous in the sense in which we have before used the word, i. e., uniform in every respect throughout its whole extent. Let <Sj, S.^., , . . >S„ be the ultiiiiate components of the fluid ; we may then consider every body which can be formed out of the fluid to be composed of S^, S2, . . . aS„, and that in only one way. Let m^, m^, . . . m„ denote the quantities of these substances in any such body, and let f, ?/, v, denote its energy, entropy, and volume. The fundamental equation for compounds of iS^, ^.Sg, . . . S„, if completely determined, will give us all possible sets of simultaneous values of these variables for homo- geneous bodies. J. W. Gihbs—Equilibriu7n of Heterogeneous Substances. 157 Now, if it is possible to assign such values to the constants T, F, M^, J/2, . . . 3f„ that the value of the expression ^ - T,/-\-Pv ~ J/, m, - 3/2 W2 . . . - 3f„m„ (133) shall be zero for the given fluid, and shall be positive for every other phase of the same ' components, i. e., for every homogeneous body* not identical in nature and state with the given fluid (but composed entirely oi S^, S^, . . . /S„), the condition of the given fluid will be stable. For, in any condition whatever of the given mass, whether or not homogeneous, or fluid, if the value of the expression (133) is not negative for any homogeneous part of the mass, its value for the whole mass cannot be negative ; and if its value cannot be zero for any homogeneous part which is not identical in phase with the mass in its given condition, its value cannot be zero for the whole except when the whole is in the given condition. Therefore, in the case supposed, the value of this expression for any other than the given condition of the mass is positive. (That this conclusion cannot be invalidated by the fact that it is not entirely correct to regard a composite mass as made up of homogeneous parts having the same properties in respect to energy, entropy, etc., as if they were parts of larger homogeneous masses, will easily appear from consider- ations similar to those adduced on pages 131-133.) If, then, the value of the expression (133) for the mass considered is less when it is in the given condition than when it is in any other, the energy of the mass in its given condition must be less than in any other condi- tion in which it has the same entropy and volume. The given con- dition is therefore stable. (See page 110.) Again, if it is possible to assign such values to the constants in (133) that the value of the expression shall be zero for the given fluid mass, and shall not be negative for any phase of the same com- ponents, the given condition will be evidently not unstable. (See page 110.) It will be stable unless it is possible for the given matter in the given volume and with the given entropy to consist of homo- geneous parts for all of which the value of the expression (133) is zero, but which are not all identical in phase with the mass in its given con- dition. (A mass consisting of such parts would be in equilibrium, as we have already seen on pages 133, 134.) In this case, if we disre- gard the quantities connected with the surfaces which divide the * A vacuum is throughout this discussion to be regarded as a limiting case of an extremely rarified body. We may thus avoid the necessity of the specific mention of a vacimm in propositions of this kind. 158 J. W. Gihbs — Equilibrium of Heterogeneous Substances. homogeneous parts, we must regard the given condition as one of neutral equilibiium. But in regard to these homogeneous parts, which we may evidently consider to be all diiFerent phases, the fol- lowing conditions must be satisfied. (The accents distinguish the letters referring to the different parts, and the unaccented letters refer to the whole mass.) Tf' + jf + etc. = ;^, 1 v' -f v" -j- etc. =: y, I m/ + m/' + etc. = w,, 1^ (134) //?2' + ''*3 " + etc. = ^2, j etc. J Now the values of //, y, m^, m^, etc. are determined by the whole fluid mass in its given state, and the values of -„ —„ etc., —r, — ^, etc — — -^ etc., etc., are determined by the phases of the various parts. But the phases of these parts are evidently determined by the phase of the fluid as given. They form, in fact, the whole set of coexistent phases of which the latter is one. Hence, we may regard (134) as n + 2 linear equations between ?>', u", etc. (The values of v' v" etc. are also subject to the condition that none of them can be negative.) Now one solution of these equations must give us the given condition of the fluid ; and it is not to be expected that they will be capable of any other solution, unless the number of different homogeneous parts, that is, the number of different coexistent phases, is o-reater than w + 2. We have already seen (page 153) that it is not probable that this is ever the case. We may, however, remark that in a certain sense an infinitely large fluid mass will be in neutral equilibrium in regard to the formation of the substances, if such there are, other than the given fluid, for which the value of (133) is zero (when the constants are so deter- mined that the value of the expression is zero for the given fluid, and not negative for any substance) ; for the tendency of such a for- mation to be reabsorbed will diminish indefinitely as the mass out of which it is formed increases. When the substances aS'j, aS^, . . . S„ are all independently vari- able components of the given mass, it is evident from (86) that the conditions that the value of (133) shall be zero for the mass as given, and shall not be negative for any phase of the same components, can only be fulfilled when the constants T, P, M^, J/g, . . . M„ are equal to the temperature, the pressure, and the several potentials in the given J. W. Gibhs — Equilibrium of Heterogeneous Substances. 159 mass. If we give these values to the constants, the expression (133) will necessarily have the value zero for the given mass and we shall only have to inquire whether its value is positive for all other phases. But when *S^j, aS^o, • . . S„ are not all independently variable compo- nents of the given mass, the values which it will he necessary to give to the constants in (133) cannot be determined entirely from the properties of the given mass ; but T and P must be equal to its temperature and pressure, and it will be easy to obtain as many equa- tions connecting J/,, J/g, . . . M„ with the potentials in the given mass as it contains independently variable components. When it is not possible to assign such values to the constants in (133) that, the value of the expression shall be zero for the given fluid, and either zero or positive for any phase of the same compo- nents, we have already seen (pages 129-134) that if equilibrium subsists without passive resistances to change, it must be in virtue of properties which are peculiar to small masses surrounded by masses of different nature, and which are not indicated by fundamental equations. In this case, the fluid will necessarily be unstable, if we extend this term to embrace all cases in which an initial disturbance confined to a small part of an indefinitely large fluid mass will cause an ultimate change of state not indefinitely small in degree through- out the whole mass. In the discussion of stability as indicated by fundamental equations it will be convenient to use the term in this sense.* * If we wish to know the stability of the given fluid when exposed to a constant tem- perature, or to a constant pressure, or to both, we have only to suppose that there is enclosed in the same envelop with the given fluid another body (which cannot combine with the fluid) of which the fundamental equation is e = Ti], or e = — Pv. or e = Ti] — Pv. as the case may be, (Tand P denoting the constant temperature and pressure, which of course must be those of the given fluid,) and to apply the criteria of page 110 to the whole system. When it is possible to assign such values to the constants in (133) that the value of the expression shall be zero for the given fluid and positive for every other phase of the same components, the value of (133) for the whole system will be less when the system is in its given condition than when it is in any other. (Changes of form and position of the given fluid are of course regarded as immaterial.) Hence the fluid is stable. When it is not possible to assign such values to the con- stants that the value of (133) shall be zero for the given fluid and zero or positive for any other phase, the fluid is of course unstable. In the remaining case, when it is possible to assign such values to the constants that the value of (133) shall be zero for the given fluid and zero or positive for every other phase, but not without the value zero for some other phase, the state of equilibrium of the fluid as stable or neutral wiU be determined by the possibility of satisfying, for any other than the given condition of the fluid, equations like (134), in which, however, the first or the second or both are to be stricken out, according as we are considering the 1 60 J. W. Gibhs — Equilihrium of Heterogeneoiis Substances. In determining for any given positive values of T and P and any- given values whatever of 3/,, M^, . . . M„ whether the expression (133) is capable of a negative value for any phase of the components aSj, aS'o, . . . /8„, and if not, whether it is capable of the value zero for any other phase than that of which the stability is in question, it is only necessary to consider phases having the temperature T and pressure P. For we may assume that a mass of matter represented by any values of m^, m^, • • • m„is capable of at least one state ot not unstable equilibrium (which may or may not be a homogeneous state) at this temperature and pressure. It may easily be shown that for such a state the value of e — T?^-^ Pv must be as small as for any other state of the same matter. The same will therefore be true of the value of (133), Therefore if this expression is capable of a negative value for any mass whatever, it will have a negative value for that mass at the temperature T and pressure P. And if this mass is not homogeneous, the value of (133) must be negative for at least one of its homogeneous parts. So also, if the expression (133) is not capable of a negative value for any phase of the comj)onents, any phase for which it has the value zero must have the temperature T and the pressure P. It may easily be shown that the same must be true in the limiting- cases in which T=.0 and P=:0. For negative values of P, (133) is always capable of negative values, as its value for a vacuum is Pv. For any body of the temperature T and pressure P, the expression (133) may by (91) be reduced to the form t, — J/i m, — 31^ m^ ... —M„m„. (135) We have already seen (pages 131, 132) that an expression like (133), when T, P, Jifj, J/g, . . . J/„ and v have any given finite values, cannot have an infinite negative value as applied to any real body. Hence, in determining whether (133) is capable of a negative value for any phase of the components aS'j, S^, . . . jS„, and if not, whether it is capable of the value zero for any other phase than that of which the stability is in question, we have only to consider the least value of which it is capable for a constant value of v. Any body giving this value must satisfy the condition that for constant volume de - T(h/ — J/, dm^ — J/^ dot^ ... — 3f„dm„^ 0, (136) stability of the fluid for constant temperature, or for constant pressure, or for both. The number of coexistent phases will sometimes exceed by one or two the number of the remaining equations, and then the equilibrium of the fluid will be neutral in respect to one or two independent changes. J. W. Gihhi< — Equilihrmia of Heterogeneous Substances. 161 or, if we substitute the value of de taken from equation (86), usinj^ sub- script a . . . g for the quantities rehiting to the actual components of the body, and subscript h . . . k for those relating to the possible, t dt] 4- //„ dm, . . . + M, dm^ -\- j.i^ dm^ . . . -+- jm dm.^ — Tdtf - 31^ dm^ — Jfs f^^'h • • • - M„dm„^ 0. (137) That is, the temperature of the body must be equal to T, and the potentials of its components must satisfy the same conditions as if it were in contact and in equilibrium with a body having potentials M^, M2, . . . M„. Therefore the same relations must subsist betAveeu //„... //,„ and M^ . . . Jf„ as between the units of the corresponding substances, so that m,/.i, . . . ■j-m^ju„ = m^ TJf^ . . . + m„ Jf„; (138) and as we have by (93) £ = t}]^p V -h IX, m„ . . . -\- pij m„ (139) the expression (133) will reduce (for the body or bodies for which it has the least value per unit of volume) to {F-p)v, (140) the value of which will be positive, null, or negative, according as the value of P — jo (141) is positive, null, or negative. Hence, the conditions in regard to the stability of a fluid of which all the ultimate components are independently variable admit a very sim- ple expression. If the pressure of the fluid is greater than that of any other phase of the same components which has the same temperature and the same values of the potentials for its actual components, the fluid is stable without coexistent phases ; if its pressure is not as great as that of some other such phase, it will be unstable ; if its pressure is as great as that of any other such phase, but not greater than that of every other, the fluid will certainly not be unstable, and in all probability it will be stable (when enclosed in a rigid envelop which is impermeable to heat and to all kinds of matter), but it will be one of a set of coexistent phases of which the others are the phases which have the same pressure. The considerations of the last two pages, by which the tests relating to the stability of a fluid are simplified, apply to such bodies as actually exist. But if we should form arbitrarily any equation as a fundamental equation, and ask whether a fluid of which the proper- Tbans. Conn. Acad., Vol. III. 21 January, 1876. 162 J. W. G-ibhs — EquUibriam of Heterogeneous Substances. ties were given by that equation would be stable, the tests of stability- last given would be insufficient, as some of our assumptions might not be fulfilled by the equation. The test, however, as first given (pages 156-159) would in all eases be sufficient. Stability in respect to Continuous Changes of Phase. In considering the changes which may take place in any mass, we have already had occasion to distinguish between infinitesimal changes in existing phases, and the formation of entirely new phases. A phase of a fluid may be stable in regard to the former kind of change, and unstable in regard to the latter. In this case it may be capable of continued existence in virtue of properties which prevent the com- mencement of discontinuous changes. But a phase which is unstable in regard to continuous changes is evidently incapable of permanent existence on a large scale except in consequence of passive resistances to change. We will now consider the conditions of stability in respect to continuous changes of phase, or, as it may also be called, stability in respect to adjacent phases. We may use the same gen- eral test as before, except that the expression (133) is to be applied only to phases which difier infinitely little from the phase of which the stability is in question. In this case the component substances to be considered will be limited to the independently variable com- ponents of the fluid, and the constants M^, M^., etc. must have the values of the potentials for these components in the given fluid. The constants in (133) are thus entirely determined and the value of the expression for the given phase is necessarily zero. If for any infi- nitely small variation of the phase, the value of (133) can become negative, the fluid will be unstable ; but if for every infinitely small variation of the phase the value of (133) becomes positive, the fluid will be stable. The only remaining case, in which the phase can be varied without altering the value of (133) can hardly be expected to occur. The phase concerned woiild in such a case have coexistent adjacent phases. It will be sufficient to discuss the condition of sta- bility (in respect to continuous changes) without coexistent adjacent phases. This condition, which for brevity's sake we Avill call the condition of stability, may be written in the form f " _ t' rf -^p'v" - fA^' m , " . . . - /V ni^' > 0, (142) in which the quantities relating to the phase of which the stability is in question are distinguished by single accents, and those relating to J. W. Gibbs — EqulUbrituti of Heterogeneous tmbstances. 1G3 the other phase by double accents. This condition is by (93) equiva- lent to 5" _ t' if +p' v" -II,' )>i," ... — //„' m„" — f' + «'?/—;/«' + /<, '/>i/ . . . -!-//„' w„'>0, (143) and to ^t'ff+pv"-,i,'m," . . . -//:»?„" 4. t" if - if v" + 1.1 ,"m^" . . . + Mn" mj' > 0. (144) The condition (143) may be expressed more briefly in the form z/f> ^ J/; — ^>z/ti + /<, z/?Hj . . . -\-/.4„Jm„, (145) if we use the character J to signify that the condition, although relating to infinitesimal differences, is not to be interpreted in accord- ance with the usual convention in respect to differential equations with neglect of infinitesimals of higher orders than the first, but is to be interpreted strictly, like an equation between finite differences. In fact, when a condition like (145) (interpreted strictly) is satisfied for infinitesimal diffei'ences, it must be possible to assign limits within which it shall hold true of finite differences. But it is to be remem- bered that the condition is not to be applied to any arbitrary values of Jyj, z/u, Zlm,, . . . Jnin, but only to such as are determined by a change of phase. (If only the quantity of the body which determines the value of the variables should vary and not its phase, the value of the first member of (145) would evidently be zero.) We may free ourselves from this limitation by making v constant, which will cause the term — p Av to disappear. If we then divide by the con- stant V, the condition will become in which form it will not be necessary to regard v as constant. As we may obtain from (86) V V V V we see that the stability of any phase in regard to continuous changes depends ujion the same conditions in regard to the second and higher differential coefficients of the density of energy regarded as a function of the density of entropy and the densities of the several components^ which would make the density of energy a minimum, if the necessary conditions in regard to the first differential coefficients were fulfilled. When //= 1, it may be more convenient to regard m as constant 164 J. W. Gihhs — Equilibrium of Heterogeneous Substances. in (145) than v. Regarding m a constant, it appears that the stability of a phase depends upon the same conditions in regard to the second and higher differential coefficients of the energy of a unit of mass regarded as a function of its entropy and volume, which would make the energy a minimum, if the necessary conditions in regard to the first differential coefficients were fulfilled. The formula (144) expresses the condition of stability for the phase to which t', p\ etc. relate. But it is evidently the necessary and sufficient condition of the stability of all phases of certain kinds of matter, or of all phases within given limits, that (144) shall hold true of any two infinitesimally diffi^ring phases within the same limits, or, as the case may be, in general. For the purpose, therefore, of such collective determinations of stability, we may neglect the distinction between the two states compared, and write the condition in the form — 1/ ^t-\-v ^p — m^ J/4^ . . . —m„JjJ„>0, (148) or Comparing (98), we see that it is necessary and sufficient for the sta- bility in regard to continuous changes of all the phases within any given limits, that within those limits the same conditions should be fulfilled in respect to the second and higher differential coefficients of the pressure regarded as a function of the temperature and the sev- eral potentials, which would make the pressure a minimum, if the necessary conditions witb i-espect to the first difierential coefficients were fiilfilled. By equations (87) and (94), the condition (142) may be brought to the form ->-?/■' ~ t' if —p' v' -\- /.ii' m^' . . . -\. ^(J m„'>0. (150) For the stability of all phases within any given limits it is necessary and sufficient that within the same limits this condition shall hold true of any two phases which differ infinitely little. This evidently requires that when v' =. d", m^' = iii ^\ . . . in„' = rnj\ f ~'/'+{t" -t'),/'>0; (151) and that when t' — t" f +P' '^" - /< 1 ' >/*i" . • . 4- /'„' mj' - f ~ P' ''' -\- M i "> i' ■ ■ ■ +/'„'/>?„' >U. (152) These conditions may be written in the form J. W. Gihhs — Equilihrium of Heterogeneous Substances. 165 [JV'4-//Z/«]„,^<0, (153) {Aip+pAv~i.i^Jm^ . . . -/<„Jw„],>0, (154) in which the subscript letters indicate the quantities which are to be resjarded as constant, m standing for all the quantities m, . . . m„. If these conditions hold true within any given limits, (150) will also hold true of any two iniinitesimally differing phases within the same limits. To prove this, we will consider a third phase, determined by the equations t"' = t', (155) and v"' = v", m/" = m,", . . . m„"' = m„". (156) Now by (153), r'-'/'"+(«"'-«") v"<o; (157) and by (154), - //'' —p'v' H-/<,'//^i' . . . 4- yU„' //<„'> 0. (158) Hghcg ' r + t" rf+p'v"'-fx,'m,"' . . . -yu„'m„"' _^/ ^t'" if -p'v' +j.{^'m,' . . . +jj„'m„'>0, (159) which by (155) and (156) is equivalent to (150). Therefore, the con- ditions (153) and (154) in respect to the phases within any given limits are necessary and sufficient for the stability of all the phases within those limits. It will be observed that in (153) we have the condition of thermal stability of a body considered as unchangeable in composition and in volume, and in (154), the condition of mechan- ical and chemical stability of the body considered as maintained at a constant temperature. Comparing equation (88), we see that the condition (153) will be satisfied, if -^ <0, i. e., if --^ or #-^ (the spe- cific heat for constant volume) is positive. When n=. 1, i. e., when the composition of the body is invariable, the condition (154) will evidently not be altered, if we regard m as constant, by which the condition will be reduced to [z/z/'-fjo J4,,„>0. (160) d^ lb dp This condition will evidently be satisfied if 3-^ ^^^ i- *?-, if -7- or _ rf^JL (the elasticity for constant temperature) is positive. But dv when 7i> 1, (154) may be abbreviated more symmetrically by making v constant. Again, by (91) and (96), the condition (142) may be brought to the form 106 J. W. Gihhs — Equillhrhim of Heterogeneous Substances. ■c^" + t" if -p" v" - fx.'m," . . . - //„' m„" -t,' -t'lf ->rp'v" +fi,'m^' . . . +//„'m„'>0. (161) Therefore, for the stability of all phases within any given limits it is necessary and sufficient that within the same limits [JC + //^« - v44„<0, (162) and [A^- fx^Am^ . . . -<-yW„Jm„],,>0, (.163) as may easily be proved by the method used with (153) and (154). The first of these formulae expresses the thermal and mechanical con- ditions of stability for a body considered as michangeable in compo- sition, and the second the conditions of chemical stability for a body considered as maintained at a constant temperature and pressure. If '/i= 1, the second condition falls away, and as in this case ? = m/<, condition (162) becomes identical with (148). The foregoing discussion will serve to illustrate the relation of the general condition of stability in regard to continuous changes to some of the principal forms of fundamental equations. It is evident that each of the conditions (146), (149), (154), (162), (163) involve in general several particular conditions of stability. We will now give our attention to the latter. Let fp ■= € — t' 7/ +p' V — ^i^' )n^ . . . — /<„'«<„, (164) the accented letters referring to one phase and the unaccented to another. It is by (142) the necessary and sufficient condition of the stability of the first phase that, for constant values of the quantities relatino- to that phase and of v, the value of $ shall be a minimiim when the second phase is identical with the first. Diflerentiating (164), we have by (86) d^ = {t - t') ch] — {p —jo') dn + (//j — /i/) dm^ ... - (Af„ - /^„')f?m„. (165) Therefore, the above condition requires that if we regard v,m^, . . . m„ as having the constant values indicated by accenting these letters, t shall be an increasing function of ;/, when the variable phase differs sufficiently little from the fixed. But as the fixed phase may be any one within the limits of stability, t must be an increasing function of // (within these limits) for any constant values of v, 'm^, . . . m,^. This condition may be written (j4J ^^- (^^^) X^ijlv, nit, . . . m„ J. W. Gibbs — Equilibrium of Heterogeneous Substances. 167 When this condition is satisfied, the value of ^, foi* any ijiven vahies oft?, wij, . . . ;/^„ will be a minimum when t-=.t'. And therefore, in applying the general condition of stability relating to the value of <^, we need only consider the phases for which t = t'. We see again by (165) that the general condition requires that if we regard ^, y, ni^., . . . m„ as having the constant values indicated by accenting these letters, //j shall be an increasing function of m,, when the variable phase difters sufficiently little from the fixed. But as the fixed phase may be any one within the limits of stability, /.i , must be an increasing function of m j (within these limits) for any constant values of ^, v, mg, . . . m„. That is, (i^) >0- (16V) When this condition is satisfied, as well as (166), ^ will have a min- imum value, for any constant values of v, m^^ . • . ?/*„, when t=it' and yu, = ///; so that in applying the general condition of stability we need only consider the phases for which t-=.t' and //j = yu/. In this way we may also obtain the follov\^ing particular conditions of stability : (4^) >0, (168) \nm^lt^ w, m,, ma, . . . ??i„ (4^\ >0. (169) \Amjt, V, mi, . . . m„_, When the 7i-\- 1 conditions (166)-(169) are all satisfied, the value of ^, for any constant value of v, will be a minimum when the tem- perature and the potentials of the variable phase are equal to those of the fixed. The pressures will then also be equal and the phases will be entirely identical. Hence, the general condition of stability will be completely satisfied, when the above particular conditions are satisfied. From the manner in which these particular conditions have been derived, it is evident that we may interchange in them a/, m^, . . . m„ in any way, provided that we also interchange in the same way ^, //,, . . . //„. In this way we may obtain different sets of n -\- 1 conditions which are necessary and sufficient for stability. The quan- tity V might be included in the first of these lists, and ~ p in the second, except in cases w^hen, in some of the phases considered, the entropy or the quantity of one of the components has the value zero. 168 J. W. Gihbs — Equilibrium of Heterogeneous Substances. Then the condition that that quantity shall be constant would create a restriction upon the variations of the phase, and cannot be substi- tuted for the condition that the volume shall be constant in the state- ment of the general condition of stability relative to the minimum value of 0. To indicate more distinctly all these particular conditions at once, we observe that the condition (144), and therefore also the condition obtained by interchanging the single and double accents, must hold true of any two infinitesimally difiering phases within the limits of stability. Combining these two conditions we have i^t" - t') [rf - rf) - {p" -p') W - ^') + (/^i" - /^i') (^i" - '-'^x) • • ' (/^"" - Z^"') «'-O>0, (170) which may be written more briefly AtAr] — ApAv-{- Jf^^Am^ . . . +J//„Jm„>0. (IVI) This must hold true of any two infinitesimally differing phases within the limits of stability. If, then, we give the value zero to one of the differences in every term except one, but not so as to make the phases completely identical, the values of the two differences in the remain- ing term will have the same sign, except in the case of Ap and Av, which will have opposite signs. (If both states are stable this will hold true even on the limits of stability.) Therefore, within the limits of stability, either of the two quantities occurring (after the sign A) in any term of (IVI) in an increasing function of the other, — except p and v, of which the opposite is true, — when we regard as constant one of the quantities occurring in each of the other terms, but not such as to make the phases identical. If we write <^? for A in (166)-(169), we obtain conditions which are always sufficient for stability. If we also substitute ^ for >, we obtain conditions which are necessary for stability. Let us consider the form which these conditions will take when ?/, v, m,, . . . m.„ are regarded as independent variables. When dv = 0, we shall have dt dt , dt ^ at=i— drj -\- - — dm , . . . + -^ — dm. dt] dm^ dm^ d\x.—^-^di]-\-^^dm. . . . + 4^dm„ [ .,^„, ^* d}] dm^ ^ dm„ \ (172) _ d^„ , , dfA^ , c?w_ , du„=z -^-dn -{--z — dm, . . . -\--~^dm„ dt] dm^^ ^ dm„ J. W. Gibbs — Equilibrium of Heterogeneous Substances. 169 Let us write i?„+i for the determinant of the order n + 1 d'^e d'-e d-^e dtf dm^ dr] dm^drj d-^e d^e d^e dij dm J dm^^ diii^dm d^e d^E d^e d)] dm^ dm. dm„ dm„^ (173) of which the constituents are by (86) the same as the coefficients in equations (1*72), and i?„, B^_-^, etc. for the minors obtained by erasing the hist column and row in the original determinant and in the minors successively obtained, and R^ for the last remaining constituent. Then if dt, dju^, . . . djn„-i, and dv all have the value zero, we have by (172) Ji„ dj.i„ = i?„+i drji„, (174) that is. /d/Jr, \ \dinjt, v,/x,, /""— 1 In like manner we obtain (d^„_i\ \dm„_Jt,v,fii, /"»-2> »»„ RZ. (175) (176) etc. Therefore, the conditions obtained by writing d for A in (166)-(169) are equivalent to this, that the determinant given above with the n minors obtained from it as above mentioned and the last remaining d^ £ constituent -y— shall all be positive. Any phase for which this con- dition is satisfied will be stable, and no phase will be stable for which any of these quantities has a negative value. But the condi- tions (166)-(169) will remain valid, if we interchange in any way 77, w^i, . . . m„ (with corresponding interchange of t, ^t^, . . . /.i„). Hence the order in which we erase successive columns with the cor- responding rows in the determinant is immaterial. Therefore none of the minors of the determinant (173) which are formed by erasing corresponding rows and columns, and none of the constituents of the principal diagonal, can be negative for a stable phase. We will now consider the conditions Avhich characterize the limits of stability (i. e., the limits which divide stable from unstable phases) Trans. Conn. Acad., Vol. III. 22 January, 1876. IVO J. W. Gibhs — EquiUhrium of Heterogeneous Substances. with respect to continuous changes.* Here, evidently, one of the conditions (166)-(169) must cease to hold true. Therefore, one of the differential coefficients formed by changing J into d in the first mem- bers of these conditions must have the value zero. (That it is the numerator and not the denominator in the differential coefficient which vanishes at the limit appears from the consideration that the denominator is in each case the differential of a quantity which is necessarily capable of progressive variation, so long at least as the phase is capable of variation at all under the conditions expressed by the subscript letters.) The same will hold true of the set of dif- ferential coefficients obtained from these by interchanging in any way rj, m^, . . . m„, and simultaneously interchanging t, j.i^, . . . /J„ in the same way. But we may obtain a more definite result than this. Let us give to rj or t, to m^ or j.i^, .. . to m„_j or /y„_i, and to v, the constant values indicated by these letters when accented. Then by (165) d^=iMu - l<)dm,. {Ill) Now ""-"•'=(,17.) '('"•-'"•') (^'«> approximately, the differential coefficient being interpreted in accord- ance with the above assignment of constant values to certain vari- ables, and its value being determined for the phase to which the accented letters refer. Therefore, and d^ = 1^^] {m„ - m„') dm,,, (179) ^ = -m^y(m„-m„')^. (180) The quantities neglected in the last equation are evidently of the same order as (v;?„ — w^„')^. Now this value of ^ will of course be different (the differential coefficient having a different meaning) according as we have made // or t constant, and according as we have made m^ or /^^ constant, etc. ; but since, within the limits of stability, the value of <?, for any constant values of «?„ and ?j, Avill be the least when t^p, 1^1 . . . //„_i have the values indicated by accenting these letters, the value of the differential coefficient will be at least as small * The limits of stability with respect to discontinuous changes are formed by phases which are coexistent with other phases. Some of the properties of such phases have already been considered. See pages 152-156. J. W. Gibhs — Equilihrmm of Heterogeneous l^ubstuHces. 171 when we give these variables these constant values, as when we adopt any other of the suppositions mentioned above in re<^ard to the quantities remaining constant. And in all these relations we may- interchange in any way //, >«,, . . . «?„, if we intercliange in the same way t, p(^, . . . i.i„. It follows that, within the limits of stability, when we choose for anj^ one of the differential coefficients dt dii J c///„ d7f dw^; ' ' ' dm„ (^^1) the quantities following the sign d in the numerators of the others together with v as those which are to remain constant in diiferentia- tion, the value of the differential coefficient as thus determined will be at least as small as when one or more of the constants in differen- tiation are taken from the denominators, one being still taken from each fraction, and v as before being constant. Now we have seen that none of these differential coefficients, as determined in any of these ways, can have a negative value within the limit of stability, and that some of them must have the value zero at that limit. Therefore, in virtue of the relations just established one at least of these differential coefficients determined by considerino- constant the quantities occurring in the numeratoi-s of the others together with v, will have the value zero. But if one such has the value zero, all such will in general have the same value. For if for example, has the value zero, we may change the density of the component S„ without altering (if we disregard infinitesimals of higher orders than the first) the temperature or the potentials, and therefore, by (98), without altering the pressure. That is, we may change the phase without altering any of the quantities t,j), /<j, , . . /,/„, (In other words, the phases adjacent to the limits of stability exhibit apj^roncimateli/ the relations characteristic of neutral equili- brium.) Now this change of phase, which changes the density of one of the components, will in general change the density of the others and the density of entropy. Therefore, all the other differen- tial coefficients formed after the analogy of (182), i, e., formed from the fractions in (181) by taking as constants for each the quantities in the numerators of the others together with u, will in general have the value zero at the limit of stability. And the relation which characterizes the limit of stability may be expressed, in general, by setting any one of these differential coefficients equal to zero. Such 172 J. W. Gibbs — Equilibrium of Heterogenous Substances. an equation, when tlie fundamental eqnation is known, may be reduced to the form of an equation between the independent variables of the fundamental equation. Again, as the determinant (IVS) is equal to the product of the differential coefficients obtained by writing d for A in the first members of (166)-(169), the equation of the limit of stability may be expressed by setting this determinant equal to zero. The form of the differential equation as thus expressed will not be altered by the interchange of the expressions ;/, «?.j, . . . »?„, but it will be altered by the substitution of v for any one of these expressions, which will be allowable whenever the quantity for which it is substituted has not the value zero in any of the phases to which the formula is to be applied. The condition formed by setting the expression (182) equal to zero is evidently equivalent to this, that that is, that 3=0, (183) or by (98), if we regard ^, //j, ... /^„ as the independent variables, (It?) = '"^ <'««> In like manner we may obtain (186) d^p d^p d^p ^-"' ^? = "'- • • diAZ7' = '^- Any one of these equations, (185), (186), may be regarded, in gen- eral, as the equation of the limit of stability. We may be certain that at every phase at that limit one at least of these equations will hold true. GEOMETRICAL ILLITSTRATIONS. Surfaces in tchich the Composition of the Body represented is Constant. In vol. ii, p. 382, of the Trans. Conn. Acad., a method is described of representing the thermodynamic properties of substances of invariable composition by means of surf^xces. The volume, entropy, and energy J. ^V. (xibhs — Equilibrium of Heterogeneous ISuhsfances. 17;5 of a constant quantity of a substance are represented by rectangular co-ordinates. This method corresponds to the first kind of limda- raental equation described on pages 140-144. Any other kind of fundamental equation for a substance of invariable composition will suggest an analogous geometrical method. Thus, if we make m con- stant, the variables in any one of the sets (99)-(103) are reduced to three, which may be represented by rectangular co-ordinates. This will, however, afford but four different methods, for, as has already (page 150) been observed, the two last sets are essentially equivalent when n ■= \. The method described in the preceding volume has certain advan- tages, especially for the purposes of theoretical discussion, but it may often be more advantageous to select a method in which the proper- ties represented by tM'o of the co-ordinates shall be such as best serve to identify and describe the different states of the substance. This condition is satisfied by temperature and pressiire as well, perhaps, as by any other properties. We may represent these by two of the co-ordinates and the potential by the third. (See page 143.) It will not be overlooked that there is the closest analogy between these three quantities in respect to their parts in the general theory of equilibrium. ( A similar analogy exists between volume, entropy, and energy.) If we give m the constant value unity, the third co-ordinate will also represent C, which then becomes equal to /<. Comparing the two methods, we observe that in one v = x, i] — y, € = z, (187) dz dz ^ dz dz ,^^^ and in the other t-z.x^ p=-y, i.i^'C,z=.z, (189) dz dz dz dz , ^ uX clx Now ^— and ^— are evidently determined by the inclination of the dx dy (Txi (XX tangent plane, and z — -^ x — -^y is the segment which it cuts ofi" on the axis of Z. The two methods, therefore, have this reciprocal relation, that the quantities represented in one by the position of a point in a surface are represented in the other by the position of a tangent plane. 1 74 J. W. Glbbs — EqmUhriuiin of Heterogeneous Substances. The surfaces detined by equations (187) and (189) may be distin- guished as the v-7]-e surface, and the t-2>'C surface, of the substance to which they relate. In the t-p-'C surface a line in which one part of the surface cuts another represents a series of pairs of coexistent states. A point through whicli pass three different parts of the surface represents a triad of coexistent states. Through such a point will evidently pass the three lines formed by the intersection of these sheets taken two by two. The perpendicular projection of these lines upon the i>t plane will give the curves which have recently been discussed by Pro- fessor J. Thomson.* These curves divide the space about the projec- tion of the triple point into six parts which may be distinguished as follows : Let C^'^^, C^^', ^^*-' denote the three ordinates determined for the same values of p and t by the three sheets passing through the triple point, then in one of the six spaces ^(n<Ki)<^(s,^ (191) in the next space, separated from the fornier by the line for which ^(n<^(S)<^(z)^ (192) in the third space, separated from the last by the line for which ^(Sj<^(n^^W (193) in the fourth ?(«> < ?(^) < C^''\ (194) in the fifth C^^> < tS^^ < ?(^), (195) in the sixth C<^> < ?(^) < ?(«>. (196) The sheet which gives the least values of 'C, is in each case that which represents the stable states of the substance. From this it is evident that in passing around the projection of the triple point we pass through lines representing alternately coexistent stable and coexistent unstable states. But the states represented by the intermediate values of ? may be called stable relatively to the states represented by the highest. The differences C^^^ — ^^'>, etc. represent the amount of woi"k obtained in bringing the substance by a reversible process from one to the other of the states to which these quantities relate, in a medium having the temperature and pressure common to the two states. To illustrate such a process, we may suppose a plane perpendicular to the axis of temperature to pass through the points * See the Keports of the British Association for 1871 and 1872 ; and Philosophical Magazine, vol. xlvii. (1874), p. 447. J. W. Gibhs—Equilibrlaui of Ileterofjeneous SuhsUmres. 175 representing tlie two states. This will in genorul cut tlie double line formed by the two sheets to which the symbols [L) and ( T^) refer. The intersections of the plane with the two sheets will connect the double point thus determined with the i)oints representino- the initial and linal states of the process, and thus form a reversible path for the body between those states. The geometrical relations which indicate tlie stability of any state may be easily obtained by applying the principles stated on pp. 156 ff. to the case in which there is but a single component. The expres- sion (133) as a test of stability will reduce to e -t'T/-\-p'v - /.I'm, (197) the accented letters referring to the state of which the stability is in question, and the unaccented letters to any other state. If we con- sider the quantity of matter in each state to be unity, this expression may be reduced by equations (91) and (96) to the form ^-l''+(«-0v-(7^-/>V, (198) which evidently denotes the distance of the point {t',p', t') below the tangent plane for the point {t, p, t), measured parallel to the axis of 'Q. Hence if the tangent plane for every other state passes above the point representing any given state, the latter will be stable. If any of the tangent planes pass below the point rejjresenting the given state, that state will be unstable. Yet it is not always necessary to consider these tangent planes. For, as has been observed on page- 160, we may assume that (in the case of any real substance) there will be at least one not unstable state for any given temperature and pressure, except when the latter is negative. Therefore the state represented by a point in the surface on the positive side of the plane jo= 0 will be unstable only when there is a point in the surface for which t and p have the same values and C a less value. It follows from what has been stated, that where the surface is doubly convex upwards (in the direction in which 'C is measured) the states repre- sented will be stable in respect to adjacent states. This also appears directly from (162). But where the surface is concave upwards in either of its principal curvatures the states represented will be unsta- ble in respect to adjacent states. When the number of component substances is greater than unity, it is not possible to represent the fundamental equation by a single surface. We have therefore to consider how it may be represented by an infinite number of surfaces. A natural extension of either of the methods already described will give us a series of surfaces in 176 J. W. Gibhs — Equilibrium of Heterogeneous Substances. which every one is the v-7]-e surface, or every one the t-p-l surface for a body of constant composition, the proportion of the components varying as we pass from one surface to another. But for a simultaneous view of the properties which are exhibited by compounds of two or three comj^onents without change of temperature or pressure, we may more advantageously make one or both of the quantities t or p con- stant in each surface. Surfaces and Curves in tchich the Composition of the Body repre- sented is Variable and its Temperature and Pressure are Constant. When there are three components, the position of a point in the J^I^plane may indicate the composition of a body most simply, per- haps, as follows. The body is supposed to be composed of the quan- tities ?7«j, //ig, i^a '^^ tlie substances ^S*,, /S'g, S^^ the value of m^ -(" mg + mg being unity. Let Pj, P^, P3 be any three points in the plane, which are not in the same straight line. If we suppose masses equal to m^, mg, m^ to be placed at these three points, the center of gravity of these masses will determine a point which will indicate the value of these quantities. If the triangle is equiangular and has the height unity, the distances of the point from the three sides will be equal numerically to «?j, m,, m^. Now if for every possible phase of the components, of a given temperature and pressure, we lav off from the point in the X- Y plane which represents the compo- sition of the phase a distance measured parallel to the axis of Z and representing the value of C (when ni^-\-n)2-\-'mQ=.\), the points thus determined will form a surface, which may be designated as the mj-mg-^Vg-C surface of the substances considered, or simply as their m-t, surface, for the given temperature and pressure. In like manner, when there are but two component substances, we may obtain a curve, which we will suppose in the X-Z plane. The coordinate y may then represent temperature or pressure. But we will limit our- selves to the consideration of the properties of the m-X, surface for w =r 3, or the m-l curve for n =z 2, regarded as a surface, or curve, which varies with the temperature and pressure. As by (96) and (92) and (for constant temperature and pressure) d'Q = f.1^ dm J -f- yWg ^^'^2 + /^3 dm^, if we imagine a tangent plane for the point to which these letters relate, and denote by l' the ordinate for any point in the plane, J. W. Gihbs — Equilibrium oj' Heterogeneous /Substances. 177 and by >«,', wig'j "^a'j ^^^^ distances of the foot of this ordinate from the three sides of the triangle Pj P3 Pg, we may easily obtain C' = /(,mj' + /-/o Wo' + Ms "'3', (199) which we may regard as the equation of the tangent plane. There- fore the ordinates for this plane at P^, P,, and P3 are equal respect- ively to the potentials yu,, yUg? 'i^^*^ /'s- -"^nd in general, the ordinate for any point in the tangent plane is equal to the potential (in the phase represented by the point of contact) for a substance of which the composition is indicated by the position of the ordinate. (See page 149.) Among the bodies which may be formed of S^, aS^, and -83, there may be some which are incapable of variation in composi- tion, or which are capable only of a single kind of variation. These will be represented by single points and curves in vertical planes. Of the tangent plane to one of these curves only a single line will be fixed, which will determine a series of potentials of which only two will be independent. The phase represented by a separate point will determine only a single potential, viz., the potential for the substance of the body itself, which will be equal to 'C. The points representing a set of coexistent phases have in general a common tangent plane. But when one of these points is situated on the edge where a sheet of the surface terminates, it is sufficient if the plane is tangent to the edge and passes below the surface. Or, when the point is at the end of a separate line belonging to the sur- face, or at an angle in the edge of a sheet, it is sufficient if the plane pass through the point and below the line or sheet. If no part of the surface lies below the tangent plane, the points where it meets the plane will represent a stable (or at least not unstable) set of co- existent phases. The surface which we have considered represents the relation between 'C, and m^, m^, m„ for homogeneous bodies when t and jo have any constant values and ni^ -|- m^ -f-^s = 1- It will often be useful to consider the surface which represents the relation between the same variables for bodies which consist of parts in different but coexistent phases. We may suppose that these are stable, at least in regard to adjacent phases, as otherwise the case would be devoid of interest. The point which represents the state of the composite body will evidently be at the center of gravity of masses equal to the parts of the body placed at the points representing the phases of these parts. Hence from the surface representing the properties of homogeneous bodies, which may be called the primitive surface, we Trans. Conn. Acad., Vol. III. 23 January, 1876. 178 tT. W. Gibhs — Equilibrium of Heterogeneous Substances. may easily construct the surface representing the properties of bodies which ai-e in equilibrium but not homogeneous. This may be called the secondary or derived surface. It will consist, in general, of various portions or sheets. The sheets which represent a combina- tion of two phases may be formed by rolling a double tangent plane upon the primitive surface : the part of the envelop of its successive positions which lies between the curves traced by the points of con- tact will belong to the derived surface. When the primitive surface has a triple tangent plane or one of higher order, the triangle in the tangent plane formed by joining the points of contact, or the smallest polygon without re-entrant angles which includes all the points of contact, will belong to the derived surface, and will represent masses consisting in general of three or more phases. Of the whole thermodynamic surface as thus constructed for any temperature and any positive pressure, that part is especially impor- tant which gives the least value of !: for any given values of Wj, ?«2? m^. The state of a mass represented by a point in this part of the surface is one in which no dissipation of energy would be possible if the mass were enclosed in a i-igid envelop impermeable both to matter and to heat ; and the state of any mass composed of aS^, aSj, S^ in any proportions, in which the dissipation of energy has been com- pleted, so far as internal processes are concerned, (i. e., under the limitations imposed by such an envelop as above supposed,) would be represented by a point in the part which we are considering of the in-'Q surface for the temperature and pressure of the mass. We may therefore briefly distinguish this part of the surface as the surface of dissipated energy. It is evident that it forms a continuous sheet, the projection of which upon the X- Y plane coincides with the triangle Pj P2 P3, (except when the pressure for which the m-? surface is constructed is negative, in which case there is no surface of dissipated energy,) that it nowhere has any convexity upward, and that the states which it represents are in no case unstable. The general properties of the m-t, lines for two component sub- stances are so similar as not to require separate consideration. We now proceed to illustrate the use of both the surfaces and the lines by the discussion of several particular cases. Three coexistent phases of two component substances may be represented by the points A, B, and C, in figure ], in which I is measured toward the top of the page from PjPg, '" , toward the left from P2Q2, and m^ toward the right from P,Qi. It is supposed that P1P2 = 1- Portions of the curves to which these points belong J. W. Gibhs — .Equilihriiim of Heterogeneous Sithstances. 179 are seen in the figure, and will be denoted by the symbols (A), (B), (C). We may, for convenience, speak of these as separate curves, without implying anything in regard to their possible continuity in parts of the diagram remote from their common tangent AC. The line of dissipated energy includes the straight line AC and portions of the primitive curves (A) and (C). Let us first consider how the diagram will be altered, if the temper- ature is varied while the pressure re- mains constant. If the temperature receives the increment dt, an ordinate of which the position is fixed will 'd'Q^ Q. b Fig. 1. P. receive the increment ( -^ 1 dt, or \dt I p^ m — // dt. (The reader will easily con- vince himself that this is true of the ordinates. for the secondary line AC, as well as of the ordinates for the primitive curves.) Now if we denote by ;/' the entropy of the phase represented by the point B considered as belonging to the curve (B), and by rf the entropy of the composite state of the same matter represented by the point B considered as belonging to the tangent to the curves (A) and (C), t (?/' — //') will denote the heat yielded by a unit of matter in passing from the first to the second of these states. If this quantity is positive, an elevation of temper- ature will evidently cause a part of the curve (B) to protrude below the tangent to (A) and (C), which will no longer form a part of the line of dissipated energy. This line will then include portions of the three curves (A), (B), and (C), and of the tangents to (A) and (B) and to (B) and (C). On the other hand, a lowering of the tempera- ture will cause the curve (B) to lie entirely above the tangent to (A) and (C), so that all the phases of the sort represented by (B) will be unstable. If t {i/ — ;/") is negative, these efl:ects will be produced by the opposite changes of temperature. The effect of a change of pressure while the temperature remains constant may be found in a manner entirely analogous. The varia- dp or V dp. Therefore, if the tion of any ordinate will be ( ^ ^ \dplt,;,i volume of the homogeneous phase represented by the point B is a greater than the volume of the same matter divided betAveen the the phases represented by A and C, an increase of pressure will give diagi'am indicating that all phases of the sort represented by curve (B) are unstable, and a decrease of pressure will give a diagram indi- 180 J. W. Gibbs — Equilibruim of Heterogeneous Substances. eating two stable pairs of coexistent phases, in each of which one of the pliases is of the sort represented by the curve (B). When the relation of the volumes is the reverse of that supposed, these results will be produced by the opposite changes of pressure. When we have four coexistent phases of three component substances, there are two cases which must be distinguished. In the iirst, one of the points of contact of the primitive surface with the qiaadruple tangent plane lies within the triangle formed by joining the other three ; in the second, the four points may be joined so as to form a quadrilateral without re-entrant angles. Figure 2 repi-esents the projection upon the A'^ Y plane (in which ni^, m^, m^ are measured) of a part of the snrftice of dissipated energy, when one of the points of contact D falls within the triangle formed by the other thi-ee A, B, 0. This surface includes the triangle ABC in the quadruple tangent plane, portions of the three sheets of the primitive surface which touch the triangle at its vertices, EAF, GBH, ICK, and portions of the three developable surfaces formed by a tangent plane rolling upon each pair of these sheets. These developable surfaces are repre- FlG. 2. sented in the figure by ruled surfaces, the lines indicating the direc- tion of their rectilinear elements. A point within the triangle ABC represents a mass of which the matter is divided, in general, between three or four different phases, in a manner not entirely determined by the position of a point. (The quantities of matter in these phases are such that if placed at the cori-esponding points. A, B, C, D, their center of gravity would be at the point representing the total mass.) J. W. Gihbs — Equilibrium of Heterogeneous Substances. 181 Such a mass, if exposed to constant temperature and pressure, would be in neutral equilibrium. A point in the developable surfaces repre- sents a mass of which the matter is divided between two coexisting phases, which are represented by the extremities of the line iu the figure passing through that point. A point in the primitive surface rejjresents of course a homogeneous mass. To determine the eftect of a change of temperature without change of pressure upon the general features of the surface of dissipated energy, we must know whether heat is absorbed or yielded by a mass in passing from the phase represented by the point D in the primitive surface to the composite state consisting of the phases A, B, and C which is represented by the same point. If the first is the case, an increase of temperature will cause the sheet (D) (i. e., the sheet of the primitive surface to which the point D belongs) to sep- ai-ate from the plane tangent to the three other sheets, so as to be situated entirely above it, and a decrease of temperature, will cause a part of the sheet (D) to protrude through the plane tangent to the other sheets. These effects will be produced by the opposite changes of temperature, when heat is yielded by a mass passing from the homogeneous to the composite state above mentioned. In like manner, to determine the effect of a vai-iation of pressure without change of temperature, we must know whether the volume for the homogeneous phase represented by D is greater or less than the volume of the same matter divided between the phases A, B, and C. If the homogeneous phase has the greater volume, an increase of pressure will cause the sheet (D) to separate from the plane tangent to the other sheets, and a diminution of pressure will cause a pai't of the sheet (D) to protrude below that tangent plane. And these effects will be produced by the opposite changes of pressure, if the homoge- neous phase has the less volume. All this appears from precisely the same considerations which were used in the analogous case for two component substances. Now when the sheet (D) rises above the plane tangent to the other sheets, the general features of the surface of dissipated energy are not altered, except by the disappearance of the point D. But when the sheet (D) protrudes below the plane tangent to the other sheets, the surface of dissipated energy will take the form indicated in figure 3. It will include portions of the four sheets of the primitive sui-face, portions of the six developable surfaces formed by a double tangent plane rollino- upon these sheets taken two by two, and portions of three triple tangent planes for these sheets taken by threes, the sheet (D) being always one of the three. 182 -T. W. Gibhs — Equilibrium of Heterogeneous Substances. But when the points of contact with the quadruple tangent plane which represent the four coexistent phases can be joined so as to form a quadrilateral ABCD (fig. 4) without reentrant angles, the surface of dissipated energy will include this plane quadrilateral, portions of the four sheets of the primitive surface which are tangent to it, and portions of the four developable surfaces formed by double Fig. 4. Fig. 5. tangent planes rolling upon the four pairs of these sheets which corres- pond to the four sides of the quadrilateral. To determine the gen- eral eifect of a variation of temperature upon the surface of dissipated energy, let us consider the composite states represented by the point I at the intersection of the diagonals of the quadrilateral. Among these states (which all relate to the same kind and quantity of matter) there is one which is composed of the phases A and C, and another which is composed of the phases B and D. Now if the entropy of the first of these states is greater than that of the second, (i. e., if heat is given ovit by a body in passing from the first to the second state at constant temperature and pi'essure,) which we may suppose without loss of generality, an elevation of temperature while the pi'essure remains constant will cause the triple tangent planes to (B), (D), and (A), and to (B), (D), and (C), to rise above the triple tangent planes to (A), (C), and (B), and to (A), (C), and (D), in the vicinity of the point I. The surface of dissipated energy will therefore take the form indicated in figure 5, in which there are two plane triangles and five developable surfaces besides portions of the four primitive sheets. A diminution of temperature wall give a different but entirely analogous form to the surface of dis- sipated energy. The quadrilateral ABCD will in this case break into two triangles along the diameter BD. The effects produced by J. TF. Gibhs — Equilibrmm of Heterogeneoxis Substances. 183 variation of the pressure wliile the temperature remains constant will of course be similar to those described. By considering the diiference of volume instead of the difference of entropy of the two states repi-e- sented by the point I in the quadruple tangent plane, we may distin- guish between the effects of increase and diminution of pressure. It should be observed that the points of contact of the quadruple tangent plane with the primitive surface may be at isolated points or curves belonging to the latter. So also, in the case of two component substances, the points of contact of the triple tangent line may be at isolated points belonging to the primitive curve. Such cases need not be separately treated, as the necessary modifications in the pre- ceding statements, when applied to such cases, are quite evident. And in the remaining discussion of this geometrical method, it will generally be left to the reader to make the necessary limitations or modificatioiis in analogoiis cases. The necessary condition in regard to simultaneous variations of temperature and pressure, in order that four coexistent phases of three components, or three coexistent phases of two components, shall remain possible, has already been deduced by purely analytical pro- cesses. (See equation (129).) We will next consider the case of two coexistent phases of identi- cal composition, and first, when the number of components is two. The coexistent phases, if each is variable in composition, will be represented by the point of contact of two curves. One of the curves will in general lie above the other except at the point of con- tact ; therefore, when the temperature and pressure remain constant, one phase cannot be varied in composition without becoming unstable, while the other phase will be stable if the proportion of either com- ponent is increased. By varying the temperature or pressure, we may cause the upper curve to protrude below the other, or to rise (relatively) entirely above it. (By comparing the volumes or the entropies of the two coexistent phases, we may easily determine which result would be produced by an increase of temperature or of pressure.) Hence, the temperatures and pressures for which two coexistent phases have the same composition form the limit to the temperatures and pressures for which such coexistent phases are pos- sible. It will be observed that as we pass this limit of temperature and pressure, the pair of coexistent phases does not simply become unstable, like pairs and triads of coexistent phases which we have considered before, but there ceases to be any such pair of coexistent phases. The same result has already been obtained analytically on 184 J. W. Gibbs — Equilibrmm of Heterogeneous Substances. page 155. But on that side of the limit on which the coexistent phases are possible, there will be two pairs of coexistent phases for the samj values of t and />, as seen in figure 6. If the curve AA' repre- sents vapor, and the curve BB' liquid, a liquid (represented by) B may exist in contact with a vapor A, and (at the same temperature and pressure) a liquid B' in contact with a vapor A', If we compare these phases in respect to their composition, we see that in one case the ^^' ^' vapor is richer than the liquid in a certain component, and in the other case poorer. Therefore, if these liquids are made to boil, the effect on their composition will be opposite. If the boiling is continued under constant pressure, the temperature will rise as the liquids approach each other in composition, and the curve BB' will rise relatively to the curve AA', until the curves are tangent to each other, when the two liquids become identical in nature, as also the vapors which they yield. In composition, and in the value of 'Q per unit of mass, the vapor will then agree with the liquid. But if the curve BB' (which has the greater curvature) represents vapor, and AA' represents liquid, the effect of boiling will make the liquids A and A' differ more in composition. In this case, the relations indi- cated in the figure will hold for a temperature higher than that for which (with tlie same pressure) the curves are tangent to one another. When two coexistent phases of three component substances have the same composition, they are represented by the point of contact of two sheets of the primitive surface. If these sheets do not intersect at the point of contact, the case is very similar to that which we have just considered. The upper sheet except at the point of contact represents unstable phases. If the temperature or pressure are so varied that a part of the upper sheet protrudes through the lower, the points of contact of a double tangent plane rolling upon the two sheets will describe a closed curve on each, and the surface of dissi- pated energy will include a portion of each sheet of the primitive sur- face united by a ring-shaped developable surface. If the sheet having the greater curvatures represents liquid, and the other sheet vapor, the boiling temperature for any given pressure will be a maximum, and the pressure of saturated vapor for any given temperature will be a minimun, when the coexistent liquid and vapor have the same composition. But if the two sheets, constructed for the temperature and pressure of the coexistent phases which have the same composition, intersect ./ W. Gihbs — Equilibrium, of Ileterogeneoiis Substances. 185 at the point of contact, the wliole primitive surface as seen from below will in general present four re-entrant furrows, radiating from the point of contact, for each of which a developable surface may he formed by a rolling double tangent plane. The diiferent parts of the surface of dissipated energy in the vicinity of the })oint of contact are represented in figure 7. ATB, ETF are parts of one sheet ot the primitive surface, and CTD, GTH are parts of the other. These are united by the developable surfaces BTC, DTE, FTG, HTA. Now we may make either sheet of the primitive surface sink relatively to the other by the pi'oper variation of temperature or pressure. If the sheet to which ATB, ETF belong is that which sinks relatively, these parts of the surface of dissipated energy will be merged in one, as well as the developable surfaces BTC, DTE, and also FTG, HTA. .(The lines CTD, BTE, ATE, HTG will separate from one another at T, each forming a continuous curve.) But if the sheet of the primitive surface which sinks relatively is that to which CTD and GTH belong, then Fig. 7. these parts will be merged in one in the sur- fiice of dissipated energy, as will be the developable surfaces BTC, ATH, and also DTE, FTG. It is evident that this is not a case of maximum or minimum tem- perature for coexistent phases under constant pressure, or of maximum or minimum pressure for coexistent phases at constant temperature. Another case of intei*est is when the composition of one of three coexistent phases is such as can be produced by combining the other two. In this case, the primitive surface must touch the same plane in three points in the same straight line. Let us distinguish the parts of the primitive surface to which these points belong as the sheets (A), (B), and (C), (C) denoting that which is intermediate in position. The sheet (C) is evidently tangent to the developable surface formed upon (A) and (B). It may or it may not intersect it at the point of contact. If it does not, it must lie above the developable sur- face, (unless it represents states which are unstable in regard to continuous changes,) and the surface of dissipated energy will include parts of the primitive sheets (A) and (B), the develop- able surface joining them, and the single point of the sheet (C) in which it meets this developable surface. Now, if the tempera- ture or pressure is varied so as to make the sheet (C) rise above the Tkans. Conn. Acad., Vol. III. 24 February, 1876. 1 86 ./ W. Gibhs—Equilibrluni of Heterogeneous Stihstances. (levelopable surface formed on the sheets (A) and (B), the surface of dissipated energy will be altered in its general features only by the removal of the single point of the sheet (C). But if the temperature or pressure is altei-ed so as to make a part of the sheet (C) protrude through the developable surface formed on (A) and (B), the surface b of dissipated energy will have the form indicated in figure 8. It will include two plane triangles ABC and A'B'C, a part of each of the sheets (A) and (B), represented in the figure by the spaces on the left of the line aAiV'a' and on the \ right of the line bBB'b', a small ^i«- 8- " part CC of the sheet (C), and de- velopable surfaces formed upon these sheets taken by pairs ACC'A', BCC'B', aABb, a'A'B'b'. the last two being dilFerent portions of the same developable surface. But if, when the primitive surface is constructed for such a temperature and pressui-e that it has three points of contact with the same plane in the same straight line, the sheet (C) (which has the middle position) at its point of contact with the triple tangent plane intersects the developable surface formed upon the other sheets (A) and (B), the surface of dissipated energy will not include this developable surface, but will consist of portions of the three primi- tive sheets with two developable surfaces formed on (A) and (C) and on (B) and (C). These developable surfaces meet one another at the point of contact of (C) with the triple tangent plane, dividing the portion of this sheet which be- c 7 longs to the surface of dissipated energy into two parts. If now the temperature or pressure are varied so as to make the sheet ((3) sink relatively to the de- velopable surface formed on (A) b' and (B), the only alteration in the general features of the sur- face of dissipated energy will be that the developable surfaces formed on (A) and (C) and on (B) and (C) will separate from one another, and the two parts of the sheet (C) will be merged in one. But a contrary variation of temperature or pressure will give a ./. W. Gihbs — Eqailibrlum of Hetet'oyeneoufi /Substances. 187 surface of dissipated energy such as is represented in figure (9), con- taining two plane triangles ABC, A'B'C belonging to triple tangent planes, a portion of the sheet (A) on the left of the line aA A'a', a por- tion of the sheet (B) on the right of the line bBB'b', two separate portions cCy and c'C'y' of the sheet (C), two separate portions aACc and a'A'C'c' of the developable surface formed on (A) and (C), two separate portions bBC;/ and h'B'C'y' of the developable surface formed on (B) and (C), and the portion A'ABB' of the developable surface formed on (A) and (B). From these geometrical relations it appears that (in general) the temperature of three coexistent phases is a maximum or minimum for constant pressure, and the pressure of three coexistent phases a maxi- mum or mininuim for constant temperature, when the composition of the three coexistent phases is such that one can be formed by com- bining the other two. This result has been obtained analytically on page 156. The preceding examples are amply sufficient to illustrate the use of the m-'C, surfaces and curves. The physical properties indicated by the nature of the siirface of dissipated energy have been only occa- sionally mentioned, as they are often far more distinctly indicated by the diagrams than they could be in words. It will be observed that a knowledge of the lines which divide the various different portions of the surface of dissipated energy and of the direction of the recti- linear elements of the developable surfaces, as projected upon the JC-Y' plane, without a knowledge of the form of the m-'Q surface in space, is sufficient for the determination (in respect to the quantity and composition of the resulting masses) of the combinations and separations of the substances, and of the changes in their states of aggregation, which take place when the substances are exposed to the temperature and pressure to which the projected lines relate, except so far as such transformations are prevented by passive re- sistances to change. CRITICAL PHASES. It has been ascertained by experiment that the variations of two •coexistent states of the same substance are in some cases limited in one direction by a terminal state at which the distinction of the coexistent states vanishes.* This state has been called the critical state. Analogous properties may doubtless be exhibited by com- pounds of variable composition without change of tempei-ature or * See Dr. Andrews " On the continuity of the gaseous and liquid states of matter." Phil. Trans., vol. 159, p. 575. I 88 J. IK Gibbs — Equ'dibruuii. of Heterogeneous Substances. pivssuro. For if, ;it iiiiy given tcniixTutuiv and pressure, two liquids nre ca})iil)le of forming a stable mixture in any ratio in ^ : m^ less than rt, and in any greater than A, n and h being the values of that ratio for two coexistent ))hases, while either can form a stable mixture with a third licjuid in all jtroportions, and any small quantities of the iirst and second can unite at once with a great quantity of the third to form a stable mixture, it may easily be seen that two coexistent mix- tures of the three liquids may be varied in composition, the tempera- ture and pressure remaining the same, from initial phases in each of which the (piantity of the third liquid is nothing, to a terminal phase in whicli the distinction of the two phases vanishes. In general, we may define a critical phase as one at which the dis- tinction between coexistent i>hases vanishes. We may suppose the coexistent phases to be stable in respect to continuous changes, for although I'elations in some icspects analogous might be imagined to hold true in regard to ])hases which are unstable in respect to con- tinuous changes, the discussion of siudi cases would be devoid of interest. But if the coexistent jthases and the critical phase are unstable only in respect to the possible formation of phases entirely ditferent from the critical and adjacent phases, the liability to such changes will in no respect affect the relations between the critical and adjacent jdiases, and need not be considered in a theoretical discussion of these relations, although it may prevent an experimental realiza- tion of the phases considered. For the sake of brevity, in the follow- ing discussion, ])hases in tlu^ vicinity of the critical phase will gen- erally be called stable, if they are unstable only in respect to the formation of phases entirely different from any in the vicinity of the critical phase. Let us first consider the number of independent variations of which a critical phase (while remaining such) is capable. If we denote by n the number of indejiendently variable components, a pair of coexis- tent phases will be capable of n independent variations, which may be expressed by the variations of ti of the quantities t, p, //^, //^, ...//„. If we limit these variations by giving to n — 1 of the quantities the constant values which they have for a certain critical phase, we obtain a linear* series of pairs of coexistent phases terminated by the critical phase. If we now vary infinitesimally the values of these n — l quantities, we shall have for the new set of values considered con- stant a new linear series of pairs of coexistent phases. Now for every pair of phases in the first series, there must be pairs of phases in the * This tonn is used to cliaracterize a series having a single degree of extension. ./. W. Gtbbs-~Equilibriurn of Jleterogeneous ISubstancts. 189 second series differing infinitely little from the pair in the first, and vice versa^ therefore the second series of coexistent phases must be terminated by a critical phase wliic^h differs, but differs infinitely little, from the first. We see, therefore, that if we vary arbitrarily the values of any n — 1 of the quantities <,^>», /^ j, /^g? • • • Hn-, ii« deter- mined by a critical phase, we obtain one and only one critical phase for each set of varied values ; i. c., a critical phase is capable of w— 1 independent variations. The quantities t,]>, //j, //g, • • . /^„, have the same values in two coexistent phases, but the ratios of the quantities ^/, w, m,, rti,^^. . . m„, are in general different in the two j)hases. Or, if for convenience we compare equal volumes of the two phases (which involves no loss of generality), the quantities //, mj, mg, , , . m„ will in general have different values in two coexistent phases. Aj)plying this to coexis- tent phases indefinitely near to a critical phase, we see that in the immediate vicinity of a critical phase, if the values of n of the quanti- ties t, J), /u^, //g? • • • Mn, iii'ti regarded as constant (as well as v), the variations of either of the others will be infinitely small compared with the variations of the quantities ?;, m^, rn^, . . . m„. This con- dition, which we may write in the form (-1^) =0, (200) characterizes, as we have seen on page 171, the limits which divide stable from unstable phases in respect to continuous changes. In fact, if we give to the quantities t, /j^, yUg, . . . yw„_i constant values determined by a i)air of coexistent phases, and to * a series of values increasing from the less to the greater of the values which it has in these coexistent phases, we determine a linear series of phases connecting the coexistent phases, in some part of which yu„ — since it has the same value in the two coexistent phases, but not a uniform value throughout the series (for if it had, which is theoretically im- probable, all these phases would be coexistent) — must be a decreasing function of ", or of m„, if v also is sujjposed constant. Therefore, the series must contain phases which are unstable in respect to con- tinuous changes. (See page 168.) And as such a pair of coexistent phases may be taken indefinitely near to any critical phase, the unstable jdiases (with resi)ect to continuous changes) must approach indefinitely near to this phase. 190 J. W. Gibbs — Equilibrmin of Heterogeneous Substances. Critical phases have similar pi'operties with reference to stability as determined with regard to discontinuous changes. For as every stable phase which has a coexistent phase lies upon the limit which separates stable from unstable phases, the same must be true of any stable critical phase. (The same may be said of critical phases which are unstable in regard to discontinuous changes if we leave out of account the liability to the particular kind of discontinuous change in respect to which the critical phase is unstable.) The linear series of phases determined by giving to n of the quanti- ties t,p,Mi-'M2i ' • • /'" ^^^ constant values which they have in any pair of coexistent phases consists of unstable phases in the part between the coexistent phases, but in the part beyond these phases in eithei" direction it consists of stable phases. Hence, if a critical phase is varied in such a manner ihntn of the quantities t,p, /.i^, yUg, . . . /v„ remain constant, it will remain stable in respect both to continuous and to discontinuous changes. Therefore, yu„.is an increasing function of m„ when t, v, j^i^, /.I2, • • • /'n-i have constant values determined by any critical phase. But as equation (200) holds true at the critical phase, the following conditions must also hold true at that phase : fd^/n„\ = 0, (201) \d}n„^)t, V, ^0. (202) If the sign of equality holds in the last condition, additional condi- tions, concerning the differential coefficients of higher orders, must be satisfied. Equations (200) and (201) may in general be called the equations of critical phases. It is evident that there are only two independent equations of this character, as a critical phase is capable oi n—l inde- pendent variations. We are not, however, absolutely certain that equation (200) will always be satisfied by a critical phase. For it is possible that the denominator in the fraction may vanish as well as the numerator for an infinitesimal change of phase in which the quantities indicated are constant. In such a case, we may suppose the subscript n to refer to some different component substance, or use another differen- tial coefficient of the same general form (such as are described on page 171 as characterizing the limits of stability in respect to con- tinuous changes), making the corresponding changes in (201) and (202). We may be certain that some of the formula^ thus formed will not fail. But for a perfectly rigorous method there is an ad van- J. W. Gibbs — Equilibrium of Heterogeneous Srdistances. 10] tage ill the use of ;;, y, ^jj/n,, . . . m„ as independent variables. The condition that the phase may be vai'ied without altering any of the quantities t, //,, //.,, ...//„ will then be expressed by the equation i?„+i=0, (203) in which /i„^^ denotes the same determinant as on page 169. To obtain the second equation characteristic of critical phases, we observe that as a phase which is critical cannot become unstable when \aried so that n of the quantities ^, jt), /<j, //g? • • • /'« remain constant, the differentia] of ^n^., for constant volume, viz., ^-^»+l^„-i-^^-^"+l dm . -U ^J^ dm„ dtn„ + (204) d// dm I cannot become negative when n of the equations (1V2) are satisfied. Neither can it have a positive value, for then its value might become negative by a change of sign of d?/, dm^, etc. Therefore the expres- sion (204) has the value zero, if w of the equations (172) are satisfied. This may be expressed by an equation aS=0, (205) in which S denotes a determinant in which the constituents are the same as in ^„+i, except in a single horizontal line, in which the differential coefficients in (204) are to be substituted. In whatever line this substitution is made, the equation (205), as well as (203), will hold true of every critical phase without exception. If we choose t, p, m^, m^, . . . m„ ?is independent variables, and write V for the determinant d^i dH dn (206) and V for the determinant formed from this by substituting for the constituents in any horizontal line the expressions IE, i^, . . . i^ (20V) the equations of critical phases will be Z7= 0, V— 0. (208) It results immediately from the definition of a critical phase, that an infinitesimal change in the condition of a mass in such a phase dm^^ dm^dm^ dm^.^dm^ dn dm^dm^ dH dm^'^ dH dm„_^ dm^ dH dH dH dm , dm^^ , dm2dmn-^ dm„_^^ 192 J. TF. Gibhs — EquiUbrmm of Heterogeneous Substances. may cause the mass, if it remains in a state of dissipated energy (i. e., in a state in which the dissipation of energy by internal processes is complete), to cease to be homogeneous. In this respect a critical phase resembles any phase which has a coexistent phase, but diifers from such phases in that the two parts into which the mass divides when it ceases to be homogeneous differ infinitely little from each other and from the original phase, and that neither of these parts is in general infinitely small. If we consider a change in the mass to be deter- mined by the values of dij, dv, dtn^, dm 2, . . . dw„, it is evident that the change in question Avill caiise the mass to cease to be homo- geneous whenever the expression ^f '""- % *+ '-i^' '"'•' ••■+^17 *"" <^'''> has a negative value. For if the mass should remain homogeneous, it would become imstable, as Ji„+i would become negative. Hence, in general, any change thus determined, or its reverse (determined by giving to dr/, dv, dm^, dm^, . . . dm„ the same values taken nega- tively), will cause the mass to cease to be homogeneous. The condi- tion which must be satisfied with refei'ence to dij, dv, diit^, dm^, . . . dm„, in order that neither the change indicated, nor the reverse, shall destroy the homogeneity of the mass, is expressed by equating the above expression to zero. But if we consider the change in the state of the mass (supposed to remain in a state of dissipated energy) to be determined by arbitrary values of vi-f 1 of the differentials dt, dp, f^/',, djx^, . . . dj.i„, the case will be entirely different. For, if the mass ceases to be homogeneous, it will consist of two coexistent phases, and as applied to these only n of the quantities t, p, /<,, //g, • • . yw„ will be independent. There- fore, for arbitrary variations of n+l of these quantities, the mass must in general remain homogeneous. But if, instead of supposing the mass to remain in a state of dissi- pated energy, we suppose that it remains homogeneous, it may easily be shown that to certain values of u-\-l of the above differentials there will correspond three different phases, of which one is stable with respect both to continuous and to discontinuous changes, another is stable with respect to the former and unstable with respect to the latter, and the third is unstable with respect to both. In general, however, if 91 of the quantities p, t, /a ^, /<^, . . . //„, or n arbitrary functions of these quantities, have the same constant values as at a critical phase, the linear series of phases thus deter- mined will be stable, in the vicinity of the critical phase. But if less J. W. Gibbs — Equilibrium, of Heterogeneous Substances. 193 than n of these quantities or functions of tlie same together with cer- tain of the quantities ?;, u, wij, Wj, . . . m„, or arbitrary functions of the latter quantities, have the same values as at a critical phase, so as to determine a linear series of phases, the differential of i?„+i in such a series of phases will not in general vanish at the critical phase, so that in general a part of the series will be unstable. We may illustrate these relations by considering separately the cases in which n^^\ and m = 2. If a mass of invariable composi- tion is in a critical state, we may keep its volume constant, and destroy its homogeneity by changing its entropy (i. e,, by adding or subtracting heat — probably the latter), or we may keep its entropy constant and destroy its homogeneity by changing its volume ; but if we keep its pressure constant we cannot destroy its homogeneity by any thermal action, nor if we keep its temperature constant can we destroy its homogeneity by any mechanical action. When a mass having two independently variable components is in a critical phase, and either its volume or its pressure is maintained constant, its homogeneity may be destroyed by a change of entropy or temperature. Or, if either its entropy or its temperature is main- tained constant, its homogeneity may be destroyed by a .change of volume or pressure. In both these cases it is supposed that the quantities of the components remain unchanged. But if we suppose both the temperature and the pressure to be maintained con- stant, the mass will remain homogeneous, however the proportion of the components be changed. Or, if a mass consists of two coexistent phases, one of which is a critical phase having two independently variable components, and either the temperature or the pressure of the mass is maintained constant, it will not be possible by mechanical or thermal means, or by changing the quantities of the components, to cause the critical phase to change into a pair of coexistent phases, so as to give three coexistent phases in the whole mass. The state- ments of this paragraph and of the preceding have reference only to infinitesimal changes.* * A brief abstract (which came to the author's notice after the above was in type) of a memoir by M. Duolaux, " Sur la separation des liquides melanges, etc." will be found in Comptes Bendus, vol. Ixxxi. (1875), p. 815. Trans. Conn. Acad., Vol. III. 26 February, 1876. 194 J. W. Gibbs — Equilibrinm of Heterogeneous Substances. ON THE VALUES OF THE POTENTIALS WHEN THE QUANTITY OF ONE OF THE COMPONENTS IS VERY SMALL. If Ave apply equation (97) to a homogeneous mass having two inde- pendently variable components S^ and S^, and make t, p, and m, constant, we obtain i'Ilh\ +mrp] -^0. (210) \dm2/t,p, m^ or Therefore, for ^2=0, either f^') =0, (211) /^2\ ^ ^_ (212) \dm2}t,p, 7/1, Now, whatever may be the composition of the mass considered, we may always so choose the substance S^ that the mass shall consist solely of that substance, and in respect to any other variable com- ponent S2, we shall have m2=-0. But equation (212) cannot hold true in general as thus applied. For it may easily be shown (as has been done with regard to the potential on pages 148, 149) that the value of a diiferential coefficient like that in (212) for any given mass, when the substance S^ (to which ^3 ^"^^ Ma relate) is determined, is independent of the particular substance which we may regard as the other component of the mass; so that, if equation (212) holds true when the substa.nce denoted by S^ has been so chosen that W2=0, it must hold true without sucli a restriction, Avhich cannot generally be the case. In fact, it is easy to prove directly that equation (211) will hold ti'ue of any phase which is stalile in regard to continuous changes and in which m^^^O, (/^w^g *'^ capable of negative as icell as positive values. For by (171), in any pliase having that kind of stability, //j is an increasing function of w/ j when t,p, and m.^ are regarded as constant. Hence, //j will have its greatest value when the mass con- sists wholly of aSj, i. e., when mg^rO. Therefore, if w^2 is capable of negative as well as positive values, equation (211) must hold true for rn.^ = 0. (This appears also from the geometrical representation of potentials in the m-t, curve. See page 177.) But if Wg is capable only of positive values, we can only conclude from the preceding considerations that the value of the differential coefficient in (211) cannot be positive. Nor, if we consider the physi- cal significance of this case, viz., that an increase of m.^ denotes an J. W. Gibbs — Equilibrium of Heterogeneous Substances. 195 addition to the mass in question of a substance not before contained in it, does any reason appear for supposing that this differential coeffi- cient has generally the value zero. To fix our ideas, let us suppose that S^ denotes water, and 8^ a salt (either anhydrous or any partic-. ular hydrate). The addition of the salt to water, previously in a state capable of equilibrium with vapor or with ice, will destroy the possibility of such equilibrium at the same temperature and pressure. The liquid will dissolve the ice, or condense the vapor, which is brought in contact with it under such circumstances, which shows that //j (the potential for water in the liquid mass) is diminished by the addition of the salt, when the temperature and pressure are main- tained constant. Now there seems to be no a priori reason for supposing that the ratio of this diminution of the potential for water to the quantity of the salt which is added vanishes with this quantity. We should rather expect that, for small quantities of the salt, an effect of this kind would be proportional to its cause, i. e., that the differential coefficient in (211) would have a finite negative value for an infinitesimal value of m^. That this is the case with respect to numerous watery solutions of salts is distinctly indicated by the experiments of Wtillner* on the tension of the vapor yielded by such solutions, and of Rtldorfff on the temperature at which ice is formed in them ; and unless we have experimental evidence that cases are numerous in which the contrary is true, it seems not unreasonable to assume, as a general law, that when tn^ has the value zero and is incapable of negative values, the differential coefficient in (211) will have a finite negative value, and that equation (212) will therefore hold true. But this case must be carefully distinguished from that in which m^ is capable of negative values, which also may be illus- trated by a solution of a salt in water. For tliis purpose let S^ denote a hydrate of the salt which can be ciystallized, and let S.-, denote water, and let us consider a liquid consisting entirely of 8^ and of such temperature and pressure as to be in equilibrium with crystals of S^. In such a liquid, an increase or a diminution of the quantity of water would alike cause crystals of 8^ to dissolve, which requires that the differential coefficient in (211) shall vanish at the particular phase of the liquid for which m, = 0. Let us return to the case in which m.^\^ incapable of negative values, and examine, without other restriction in regard to the substances * Fogg. Ann., vol. ciii. (1858), p. 529 ; vol. cv. (1858), p. 85; vol. ex. (1860), p. 564. \ Pogg. Ann., vol. cxiv. (1861), p. 63. 196 J. W. Gibbs — Equilibrium of Heterogeneous Substa7ices. denoted by *S'i and S^, the relation between //g and ^ tor any con- stant temperature and pressure and for such small values of -^ that the differential coefficient in (211) may be regarded as having the same constant value as when m^ = 0, the values of t, p, and m ^ being un- changed. If we denote this value of the differential coefficient by — the value of ^ will be positive, and will be independent of m^. m^ ' Then for small values of '^, we have by (210), approximately, ^2 i. e., ^\dm2/t, p, m, \rtlog rn2/t,p, Ml If we write the integral of this equation in the form pi2=Alog-^^, (215) J^ like A will have a positive value depending only upon the tempera- ture and pressure. As this equation is to be applied only to cases in which the value of m^ is very small compared with ^)t^, we may regard — - as constant, when temperature and pressure are constant, and write p(^ = A\og—^, (216) C denoting a positive quantity, dependent only upon the temperature and pressure. We have so far considered the composition of the body as varying only in regard to the proj^ortion of two comi^onents. But the argu- ment will be in no respect invalidated, if we suppose the composition of the body to be capable of other variations. In this case, the quan- tities A and 6' will be functions not only of the temperature and pressure but also of the quantities which express the composition of the substance of which together with S^ the body is composed. If the quantities of any of the components besides yS'a are very small (relatively to the quantities of others), it seems reasonable to assume that the value of ju^, and therefore the values of .1 and C, will be nearly the same as if these components were absent. J. W. Gihhs — EquUibriiim of Heterogeneous Substances. 197 Hence, if the independently variable components of any body are aS„, . . . Sg, and S/,, . . . /iS'^., the quantities of the latter being very small as compared with the quantities of the former, and are incapable of negative values, we may express approximately the values of the ])otentials for S,„ . . . /Si. by equations (subject of coui-se to the uncer- tainties of the assumptions which have been made) of the form M,.= A,\og^'f^; (217) //,=A•log-^^ (218) V in which A,^, C\, . . . A^., C^. denote functions of the temperature, the pressure, and the ratios of the quantities ni„, . . . rn^. We shall see hereafter, when we come to consider the properties of gases, that these equations may be verified experimentally in a very large class of cases, so that we have considerable reason for believing that they express a general law in regard to the limiting values of potentials.* ON CERTAIN POINTS KELATING TO THE MOLECULAR CONSTITUTION OF BODIES. It not unfrequently occurs that the number of proximate compo- nents which it is necessary to recognize as independently variable in a body exceeds the number of components which would be sufficient to express its ultimate composition. Such is the case, for example, as has been remarked on page 117, in regard to a mixture at ordinary temperatures of vapor of water and free hydrogen and oxygen. This case is explained by the existence of three sorts of molecules in the gaseous mass, viz., molecules of hydrogen, of oxygen, and of hydrogen and oxygen combined. In other cases, which are essentially the same in principle, we suppose a greater number of different sorts of molecules, which differ in composition, and the relations between * The reader will not fail to remark that, if we could assume the universality of this law, the statement of the conditions necessary for equilibrium between different masses in contact would be much simplified. For, as the potential for a substance which is only & possible component (see page 117) would always have the value — oo^ the case could not 6ccur that the potential for any substance should have a greater vakie in a mass in which that substance is only a possible component, than in another mass in which it is an actual component; and the conditions (22) and (51) might be expressed with the sign of equality without exception for the case of possible components. 198 J. W. Gibhs — Equilibrium of Heterogeneous Substances. these may be more complicated. Other cases are explained by mole- cules which differ in the quantity of matter which they contain, but not in the kind of matter, nor in the proportion of the different kinds. In still other cases, there appear to be different sorts of molecules, which differ neither in the kind nor in the quantity of matter which they contain, but only in the manner in which they are constituted. What is essential in the cases referred to is that a certain number of some sort or sorts of molecules shall be equivalent to a certain number of some other sort or sorts in respect to the kinds and quantities of matter which they collectively contain, and yet the former shall never be transformed into the latter within the body considered, nor the latter into the former, however the proportion of the numbers of the different sorts of molecules may be varied, or the composition of the body in other respects, or its thermodynamic state as represented by temperature and pressure or any other two suitable variables, pro- vided, it may be, that these variations do not exceed certain limits. Thus, in the example given above, the temperature must not be raised beyond a certain limit, or molecules of hydrogen and of oxygen may be transformed into molecules of water. The differences in bodies resulting from such differences in the con- stitution of their molecules are capable of continuous variation, in bodies containing the same matter and in the same thermodynamic state as determined, for example, by pressui-e and temperature, as the numbers of the molecules of the different sorts are varied. These differences are thus distinguished from those which depend upon the manner in which the molecules are combined to form sensible masses. The latter do not cause an increase in the number of variables in the fundamental equation ; but they may be the cause of different values of which the function is sometimes capable for one set of values of the independent variables, as, for example, when we have several different values of t, for the same values of ^, jo, m^, ni^, . . . m„, one perhaps being for a gaseous body, one for a liquid, one for an amor- phous solid, and others for different kinds of crystals, and all being invariable for constant values of the above mentioned independent variables. But it must be observed that when the differences in the constitu- tion of the molecules are entirely determined by the quantities of the different kinds of matter in a body with the two variables which express its thermodynamic state, these differences will not involve any increase in the number of variables in the fundamental equation. For example, if we should raise the temperature of the mixture of -/ W. Gibhs — EquiUhriiim of Heterogeneoiis Substcmces. 199 vapor of water and free hydrogen and oxygen, which we have just considered, to a point at which the numbers of the] different sorts of molecules are entirely determined by the temperature and pressure and the total quantities of hydrogen and of oxygen which are present, the fundamental equation of such a mass would involve but four inde- pendent variables, which might be the four quantities just mentioned. The fact of a certain part of the matter j^resent existing in the form of vapor of water would, of course, be one of the facts which determine the nature of the relation between ? and the independent variables, which is expressed by the fundamental equation. But in the case first considered, in which the quantities of the different sorts of molecules are not determined by the temperature and pressure and the quantities of the difierent kinds of matter in the body as determined by its ultimate analysis, the components of which the quantities or the potentials appear in the fimdamental equation must be those which are detei-mined by the proximate analysis of the body, so that the variations in their quantities, with two variations relating to the thermodynamic state of the body, shall include all the variations of which the body is capable.* Such cases present no especial difficulty; there is indeed nothing in the physical and chemical jiroperties of such bodies, so far as a certain range of experi- ments is concerned, Avhich is different from what might be, if the proximate components were incapable of farther reduction or trans- formation. Yet among the the various phases of the kinds of matter concerned, represented by the different sets of values of the variables which satisfy the fundamental equation, there is a certain class which merit especial attention. These are the phases for which the entropy has a maximum value for the same matter, as determined by the ultimate analysis of the body, with the same energy and volume. To fix our ideas let us call the proximate components S^, . . . S„^ and the ultimate components S„^ . . . *S/, ; and let m^, . . . m„ denote the quantities of the former, and m„, . , . m^, the quantities of the latter. It is evident that m^ . . . m^ are homogeneous functions of the first degree of m,, . . . J7^„; and that the relations between the substances aSj, . . . /8„ might be expressed by homogeneous equations of the first degree between the units of these substances, equal in number to the difference of the numbers of the proximate and of the ultimate com- * The terms proximate or ultimate are not necessarily to be understood in an abso- lute sense. All that is said here and in the following paragraphs will apply to many cases in which components may conveniently be regarded as proximate or ultimate, which are such only in a relative sense. 200 ./ W. Gihhs — Equilibrium of Heterogeneous Substances. ponents. The phases in question are those for which 7/ is a maximum for constant values of £, v, w„, . . . m,, ; or, as they may also be described, those for which e is a minimum for constant values of ?;, v, m„ . . . ni,, ; or for which 'Q is a minimum for constant values of t, p, m„, . . . m,,. The phases which satisfy this condition may be readily determined when the fundamental equation (which will con- tain the quantities m^, . . . ni„ or yWj, . . . //„,) is known. Indeed it is easy to see that we may express the conditions which determine these phases by substituting /<j, ...//„ for the letters denoting the units of the corresponding substances in the equations which express the equivalence in ultimate analysis between these units. These phases may be called, with reference to the kind of change which we are considering, phases of dissipated energ}^ That we have used a similar term before, with reference to a diiferent kind of changes, yet in a sense entirely analogous, need not create confusion. Tt is chai-acteristic of these phases that we cannot alter the values of wij, . . . Wn in any real mass in such a phase, while the volume of the mass as well as its matter remain unchanged, without diminish- in o- the energy or increasing the entropy of some other system. Hence, if the mass is large, its equilibrium can be but slightly dis- turbed by the action of any small body, or by a single electric spark, or by any cause which is not in some way proportioned to the effect to be produced. But when the proportion of the proximate compo- nents of a mass taken in connection with its temperature and pressure is not such as to constitute a phase of dissipated energy, it may be possible to cause great changes in the mass by the contact of a very small body. Indeed it is possible that the changes produced by such contact may only be limited by the attainment of a phase of dissipated energy. Such a result will probably be produced in a fluid mass by contact with another fliiid which contains molecules of all the kinds which occur in the first fluid (or at least all those which contain the same kinds of matter which also occur in other sorts of molecules), but which differs from the first fluid in that the quantities of the various kinds of molecules are entirely determined by the ultimate composition of the fluid and its temperature and pressure. Or, to speak without reference to the molecular state of the fluid, the result considered would doubtless be brought about by contact with another fluid which absorbs all the proximate components of the first, S ... aS'„, (or all those betw-ien which there exist relations of equiva- lence in respect to their ultimate analysis), independently, and with- out passive resistances, but for which the phase is completely deter- ./ W. Gibhs — Equilibrium of Heterogeneous 8ubstcm<:es. 201 mined by its temperature and pressure and its ultimate composition (in respect at least to the particular substances just mentioned). By the absorption of the substances 8^, . . . S^ independently and without passive resistances, it is meant that when the absorbing body is in equilibrium with another containing these substances, it shall be possible by infinitesimal changes in these bodies to produce the ex- change of all these substances in either direction and independently. An exception to the preceding statement may of course be made for cases in which the result in question is prevented by the occurrence of some other kinds of change ; in other words, it is assumed that the two bodies can remain in contact preserving the properties which have been mentioned. The term catalysis has been applied to such action as we are con- sidering. When a body has the property of reducing another, with- out limitation with respect to the proportion of the two bodies, to a phase of dissipated energy, in regard to a certain kind of molecular change, it may be called a perfect catalytic ar/ent with respect to the second body and the kind of molecular change considered. It seems not improbable that in some cases in which molecular changes take place slowly in homogeneous bodies, a mass of which the temperature and pressure are maintained constant will be finally brought to a state of equilibrium which is entirely determined by its temperature and pressure and the quantities of its ultimate compo- nents, while the various transitory states through which the mass passes, (which are evidently not completely defined by the quantities just mentioned,) may be completely defined by the quantities of cer- tain proximate components with the temperature and pressure, and the matter of the mass may be brought by processes approximately reversible from permanent states to these varioiis transitory states. In such cases, we may form a fundamental equation with reference to all possible phases, whether transitory or permanent; and we may also form a fundamental equation of different import and containing a smaller number of independent variables, which has reference solely to the final phases of equilibrium. The latter are the phases of dissi- pated energy (with reference to molecular changes), and when the more general form of the fundamental equation is known, it will be easy to derive from it the fundamental equation for these permanent phases alone. Now, as these relations, theoretically considered, are independent of the rapidity of the molecular changes, the question naturally arises, whether in cases in which we are not able to distinguish such trausi- Trans. Conn. Acad., Vol. III. 26 February, 1876. 202 J. W. Gibbs — Equilibrmm of Heterogeneous Substances. tory phases, they may not still have a theoretical significance. If so, the consideration of the subject from this point of view, may assist us, in such cases, in discovering the foi-m of the fundamental equation with reference to the ultimate components, which is the only equation required to express all the properties of the bodies which are capable of experimental demonstration. Thus, when the phase of a body is completely determined by the quantities of n independently vari- able components, with the temperature and pressure, and we have reason to suppose that the body is composed of a greater number n' of proximate components, which are therefore not independ- ently variable (while the temperature and pressure remain constant), it seems quite possible that the fundamental equation of the body may be of the same form as the equation for the phases of dissi- pated energy of analogous compounds of n' proximate and n ultimate components, in which the proximate components are capable of independent variation (without variation of temperature or pressure). And if such is found to be the case, the fact will be of interest as affording an indication concerning the proximate constitution of the body. Such considerations seem to be especially applicable to the very common case in which at certain temperatures and pressures, regarded as constant, the quantities of certain proximate components of a mass are capable of independent variations, and all the phases pro- duced by these variations are permanent in their nature, while at other temperatures and pressures, likewise regarded as constant, th^ quan- tities of these proximate components are not capable of independent variation, and the phase may be completely defined by the quantities of the ultimate components with the temperature and pressure. There may be, at certain intermediate temperatures and pi*essures, a condi- tion with respect to the independence of the proximate components intermediate in character, in which the quantities of the proximate components are independently variable when we consider all phases, the essentially transitory as well as the permanent, but in which these quantities are not independently variable when we consider the permanent phases alone. Now we have no reason to believe that the passing of a body in a state of dissipated energy from one to another of the three conditions mentioned has any necessary connection with any discontinuous change of state. Passing the limit which separates one of these states from another will not therefore involve any dis- continuous change in the values of any of the quantities enumerated in (99)-(103) on page 143, if >y/,, wig, . . . m„, //j, //g? • • • yWn are J. W. Gi.bbs — Equilibrium of Heterogeneous Substances. 203 understood as always relating to the ultimate components of the body. Therefore, if we regard masses in the diiferent conditions mentioned above as having different fundamental equations, (which we may sup- pose to be of any one of the five kinds described on page 143,) these equations will agree at the limits dividing these conditions not only in the values of all the variables which appear in the equations, but also in all the difi'erential coefficients of the first order involving these variables. We may illustrate these relations by supposing the values of t, />, and 'Q for a mass in which the quantities of the ultimate com- ponents are constant to be represented by rectilinear coordinates. Where the proximate composition of such a mass is not determined by t and jo, the value of I will not be determined by these variables, and the points representing connected values of t, ^>, and ^ will form a solid. This solid will be bounded in the direction opposite to that in which l is measured, by a surface which represents the phases of dissipated energy. In a part of the figure, all the phases thus repre- sented may be permanent, in another part only the phases in the bounding surface, and in a third part there may be no such solid figure (for any phases of which the existence is experimentally demonstrable), but only a surface. This surface together with the bounding surfaces representing phases of dissipated energy in the parts of the figure mentioned above forms a continuous sheet, without discontinuity in regard to the direction of its normal at the limits dividing the different parts of the figure which have been mentioned. (There may, indeed, be different sheets representing liquid and gaseous states, etc., but if we limit our consideration to states of one of these sorts, the case will be as has been stated.) We shall hereafter, in the discussion of the fundamental equations of gases, have an example of the derivation of the fundamental equa- tion for phases of dissipated energy (with respect to the molecular changes on which the proximate composition of the body depends) from the more general form of the fundamental equation. THE CONDITIONS OF EQUILIBRIUM FOR HETEROGENEOUS MASSES UNDER THE INFLUENCE OF GRAVITY. Let US now seek the conditions of equilibrium for a mass of various kinds of matter subject to the influence of gravity. It will be con- venient to suppose the mass enclosed in an immovable envelop which is impermeable to matter and to heat, and in other respects, except in regard to gravity, to make the same suppositions as on pages 115, 116. The energy of the mass will now consist of two parts, one of 204 ./. W. Gibbs — Equilihrmm of Heterogeneous Substances. which depends upon its intrinsic nature and state, and the other npon its position in space. Let Dtn denote an element of the mass, Ds the intrinsic energy of this element, h its height aboA'e a fixed horizontal plane, and g the force of gravity ; then the total energy of the mass (when without sensible motions) will be expressed by the formula fI)e-\-fghDm., (219) in which the integrations include all the elements of the mass ; and the general condition of equilibrium will be dfBe + 6fg h Dm ^ 0, (220) the variations being subject to certain equations of condition. These must express that the entropy of the whole mass is constant, that the surface bounding the whole mass is fixed, and that the total quanti- ties of each of the component substances is constant. We shall sup- pose that there are no otlier equations of condition, and that the independently variable components are the same throughout the whole mass ; and we shall at first limit ourselves to the consideration of the conditions of equilibrium with respect to the changes which may be expressed by infinitesimal variations of the quantities which define the initial state of the mass, without regarding the possibility of the formation at any place of infinitesimal masses entirely different from any initially existing in the same vicinity. Let Z>//, Dv., JJm^, . . . J)m„ denote the entropy of the element J)ni, its volume, and the quantities which it contains of the various components. Then Dm = Dm^ . . . + Dm„, (221) and dJ)m= dBm^ • • • + ^-Z>m„. (222) Also, by equation (12), 6D€ = t SDrj — ^ dUv + ju^ SJJm^ . . . -f yM„ 6J)m„. (223) By these equations the general condition of equilibrium may be reduced to the form ft SDi] ~fp 6Dv +f/i, SBm^ . . . -f ///„ dDm„ + fg 6h Biti -\-fg h 6 Dm ^ . . . -\- fgh dDm„ ^0. (224 ) Now it will be observed that the different equations of condition affect different parts of this condition, so that we must have, sepa- rately, ft 6Dt] i 0, if fSDt] = 0 ; (225) ./. W. Glbbs — Equilibrium, of Heterogeneous Substances. 205 -fp 6Bv -\-fg 6h Urn ^ 0, (220) if the bounding surface is unvaried ; y7<i 61>m^ + fgh 6Bm^ ^0, if fSDm^ = 0 ; (227) y>„ SJ}ni„ + fg h 61>m„ ^ 0, if f 6Dm^ = 0. From (225) we may derive the condition of thermal equilibrium, «z= Const. (328) Condition (226) is evidently the ordinary mechanical condition of equilibrium, and may be transformed by any of the usual methods. We may, for example, apply the formula to such motions as might take place longitudinally within an infinitely narrow tube, terminated at both ends by the external surface of the mass, but otherwise of indeterminate form. If we denote by m the mass, and by v the volume, included in the part of the tube between one end and a transverse section of variable position, the condition will take the form — fp ddv + fg Sh dm ^ 0, (229) in which the integrations include the whole contents of the tube. Since no motion is possible at the ends of the tube, fp Sdv + fdv dp =fd{p Sv) z= 0, (230) Again, if we denote by y the density of the fluid, dh fg dh dm :=fg -^ Sv y dv =.fg y Sv dh. (231 ) By these equations condition (229) may be reduced to the form fSv {dp -{- g y dh) ^ 0. (232) Therefore, since Sv is arbitrary in value, dp = — g y dh, (233) which will hold true at any point in the tube, the difierentials being taken with respect to the direction of the tube at that point. There- fore, as the form of the tube is indeterminate, this equation must hold true, without restriction, throughout the whole mass. It evi- dently requires that the pressure shall be a function of the height alone, and that the density shall be equal to the first derivative of this function, divided by — g. Conditions (227) contain all that is characteristic of chemical equilibrium. To satisfy these conditions it is necessary and sufficient that yu J -f- ^ A = Const. \ (2-34) //„ -f gh =. Const. ) 206 J. W. Glbhs — Equilibrium of Heterogeneous Substances. The expressions /^j, . . . /^„ denote quantities which we have called the potentials for the several components, and which are entirely determined at any point in a mass by the nature and state of the mass about that point. We may avoid all confusion between these quantities and the potential of the force of gravity, if we distinguish the former, when necessary, as intrinsic potentials. The relations indicated by equations (234) may then be expressed as follows : When a fluid mass is in equilibrium under the influence of gravity^ and has the same independently variable components throughout^ the intrinsic potentials for the several components are constant in any given level, and diminish uniformly as the height increases, the differ- ence of the values of the intrinsic potential for any component at two different levels, being equal to the work done by the force of gravity when a unit of matter falls from the higher to the lower level. The conditions expressed by equations (228), (233), (234) are necessary and sufficient for equilibrium, except with respect to the possible formation of masses which are not approximately identical in phase with any previously existing about the points where they may be formed. The possibility of such formations at any point is evidently independent of the action of gravity, and is determined entirely by the phase or phases of the matter about that point. The conditions of equilibrium in this respect have been discussed on pages 128-134. But equations (228), (233), and (234) are not entirely independent. For with respect to any mass in which there are no surfaces of dis- continuity (i. e., surfaces where adjacent elements of mass have finite differences of phase), one of these equations will be a consequence of the others. Thus by (228) and (234), we may obtain from (97), which will hold true of any continuous variations of phase, the equa- tion V dpz^ — g {m J . . . -f- m„) dh ; (235) or dp= - gy dh ; (236) which will therefore hold true in any mass in which equations (228) and (234) are satisfied, and in which there are no surfaces of discon- tinuity. But the condition of equilibrium expressed by equation (233) has no exception with respect to surfaces of discontinuity; therefore in any mass in which such surfaces occur, it will be necessary for equilibrium, in addition to the relations expressed by equations (228) and (234), that there shall be no discontinuous change of pressure at these surfaces. This superfluity in the particular conditions of equilibrium which we have found, as applied to a mass which is everywhere continuous •7. ]V. Gibhs — Equllihrimn of Heterogeneous Substances. 20V in phase, is due to the fact that we have made the elements of volume variable in position and size, while the matter initially contained in these elements is not supposed to be confined to them. Now, as the different components may move in different directions when the state of the system varies, it is evidently imi)ossible to define the elements of volume so as always to include the same matter ; we must, therefore, suppose the matter contained in the elements of volume to vary ; and therefore it would be allowable to make these elements fixed in space. If the given mass has no surfaces of discon- tinuity, this would be much the simplest plan. But if there are any surfaces of discontinuity, it will be possible for the state of the given mass to vary, not only by infinitesimal changes of phase in the fixed elements of volume, but also by movements of the surfaces of discon- tinuity. It would therefore be necessary to add to our general con- dition of equilibrium terms relating to discontinuous changes in the elements of volume about these surfaces, — a necessity which is avoided if we consider these elements movable, as we can then sup- pose that each element remains always on the same side of the surface of discontinuity. Method of treating the preceding jyrohlem^imiiMch the elements of volume are regarded as fixed. It may be interesting to see in detail how the particular conditions of equilibrium may be obtained if we regard the elements of volume as fixed in position and size, and consider the possibility of finite as well as infinitesimal changes of phase in each element of volume. If we use the character A to denote the differences determined by such finite differences of phase, we may express the variation of the intrin- sic energy of tlie whole mass in the form fSBe + fABe, (237) in which the first integral extends over all the elements which are infinitesimally varied, and the second over all those which experience a finite variation. We may regard both integrals as extending throughout the whole mass, but their values will be zero except for the parts mentioned. If we do not wish to limit ourselves to the consideration of masses so small that the force of gravity can be regarded as constant in direction and in intensity, we may use T to denote the potential of the force of gravity, and express the variation of the part of the energy which is due to gravity in the form -y r 6 Dm -fT ADm. (238) 20S J. W. Gihhs — EqiiUihvii(m, of Heterogeneous Substances. We shall then have, for the general condition of equilihrium, fSDe + /AUt -jTSDm - fTADm ^ 0 ; (239) and the equations of condition will be fSDi^ -\-fADt] — 0, (240) . . . ' . . . [ (241) fdl)m„ + fABrn^ = 0. ) We may obtain a condition of equilibrium independent of these equa- tions of condition, by subtracting these equations, multiplied each by an indeterminate constant, from condition (239). If we denote these indeterminate constants by T, 31^, . .. M„, we shall obtain after arranging the terms / SDs — T 6Dm — TdDtj - iHf, SDm^ . . . ^ M„ 6Dm„ fADe-TADm - TADi]-M^ ADm^ . . . -M„dI>m„^o^ (242) The variations, both infinitesimal and finite, in this condition are independent of the equations of condition (240) and (241), and are only subject to the condition that the varied values of J)e, i>//, Dm^, . . . lJm„ for each element are determined by a certain change of phase. But as we do not suppose the same element to experi- ence both a finite and an infinitesimal change of phase, we must have SJ)e~ FdDm - TdBi] - 31^ SBrn^ . . . - M„ SBm.„^0, (243) and ADs — TADm - TAD?? — M^ A Dm, ... - 3/„ JX>w„^0. (244) By equation (12), and in virtue of the necessary relation (222), the first of these conditions reduces to {t — T) dDi] + (yu, - r— J^/,) SBm^ . . . + (yu„ - r- M„) dDm„^0 ; (245) for which it is necessary and suflicient that t = r, (246) V* (247) * The gravitation potential is here supposed to be defined in the usual way. But if it were defined so as to decrease when a body falls, we would have the sign + instead of — in these equations ; i. e., for each component, the sum of the gravitation and intrinsic potentials would be constant throughout the whole mass. J. W. Gibbs — JSquilibHuni of Heteroffe)ieous Substances. 209 Condition (244) may be reduced to the form ADe^ TJDj] - {r-\-M^)JBm, ..._(]"+ ]\QJJ)m„^0; (248) and by (246) and (247) to JDe - tJDj) - //, JBm^ ... — //„ JZ>^/?„^ 0. (249) If values determined subsequently to the change of phase are distin- guished by accents, this condition may be written J)s' - t Df/ - //j Diu^' ... - /.i„Brn„' — Be + t D)i -{- 1.1^ Bm^ ... + //„ Bm^ 0, (250) which may be reduced by (93) to Be' - tB)]' - //, Bm^, ... - j.i„Bi>i„' -]- pBv^O. (251) Now if the element of volume Bv is adjacent to a surface of discon- tinuity, let us suppose Bi\ Bif, Bm^\ . . . Bm„' to be determined (for the same element of volume) by the phase existing on the other side of the surface of discontinuity. As ^, //,,.. . //„ have the same values on both sides of this surface, the condition may be reduced by (93) to — p'Bv +pBv^O. (252) That is, the pressure must not be greater on one side of a surface of discontinuity than on the other. Applied more generally, (251) expresses the condition of equilibrium with respect to the possibility of discontinuous changes of phases at any point. As Bv' = Bv, the condition may also be written Be' - tB}/ +pBij' - yt<, i>m,', ... - u„Bm„'^0, (253) which must hold true when t, p, /a^, . . . //„ have values determined by any point in the mass, and Ba', Bt/, Bv', Btn^ , . . . BmJ, have values determined by any possible phase of the substances of which the mass is composed. The application of the condition is, however, subject to the limitations considered on pages 128-134. It may easily be shown (see pages 160, 161) that for constant values of t, //,, . . . //„, and of Bv' , the first member of (253) will have the least possi- ble value when Be', Bif, Bm j ', . . . Bm^ are determined by a phase for which the temperature has the value t, and the potentials the values yt<,, . . . //„. It will be sufficient, therefore, to consider the condition as applied to such phases, in which case it may be reduced by (93) to p—p'^O. (254) That is, the pressure at any point must be as gieat as that of any phase of the same components, for which the temperature and the Trans. Conn. Acad., Vol. III. 27 April, 1876. 210 J. W. Gibhs — Equilibrium of Heterogeneous Substances. potentials have the same values as at that point. We may also express this condition by saying that the pressure nnist be as great as is consistent with equations (246), (247). This condition with the equations mentioned will always be sufficient foi- equilibrium ; when the condition is not satisfied, if equilibrium subsists, it will be at least practically unstable- Hence, the phase at any point of a fluid mass, which is in stable equilibrium under the influence of gravity (whether this force is due to external bodies or to the mass itself), and which has throughout the same independently variable components, is completely deter- mined by the phase at any other point and the difierence of the values of the gravitation potential for the two points. FUNDAMENTAL EQUATIONS OF IDEAL GASES AND GAS-MIXTUKES. For a constant quantity of a perfect or ideal gas, the product of the volume and pressure is proportional to the temperature, and the variations of energy are proportional to the variations of tempera- ture. For a unit of such a gas we may write p v:= a t^ de z=. c dt, a and c denoting constants. By integration, we obtain the equation e= ct+E, in which S also denotes a constant. If by these equations we elimin- ate t and p from (11), we obtain s-E , a £-E , de =z d?} dv, C V c or d€ , dv c vt = dv - (/ — . The integral of this equation may be written in the form c log =: // — a log V — JI, where ^denotes a fourth constant. We may regard ^as denoting the energy of a unit of the gas for ^=0 ; ^its entropy for ^=1 and v=zl ; a its pressure in the latter state, or its volume for t=l and p=zl ; c its specific heat at constant volume. We may extend the application of the equation to any quantity of the gas, without altering the values of the constants, if we substitute — , -, — for e, ri, v. respec- m m m i i-> i y tively. This will give J. W. Gibbs — B,quilibriu)n of Heterogeneous Substances. 211 1 £ — Em 7; ^_ , m , , c loar = — — H + aXocf—. (255) em in v This is a fundamental equation (see pages 140-144) for an ideal gas of invariable composition. It Avill be observed that if we do not have to consider the properties of the matter which forms the gas as ap- pearing in any other form or combination, but solely as constituting the gas in question (in a state of jjurity), we may without loss of generality give to E and H the value zero, or any other arbitrary values. But when the scope of our investigations is not thus limited, we may have determined the states of the substance of the gas for which ez=:Q and ;/=:0 with reference to some other form in which the substance appears, or, if the substance is compound, the states of its components for which ez=.0 and ;/=0 may be already determined ; so that the constants E and H cannot in general be treated as arbitrary. We obtain from (255) by differentiation ; , 1 , <x , / cE c+a f/ \ y f^ de= -dt/ dv + ( ^r- + — -2 (^ni, 256) f^m m V \e — Jlini m m^/ ' 8-E'. whence, in virtue of the general relation expressed by (86), e — Em c m (257) 8 — Em , ^ , p = a , (258) cv u = E+ —-^\c m. + a m - ?/). (259) We may obtain the fundamental equation between //•, t, i\ and ?n from equations (87), (255), and (257). Eliminating £ we have if' =z Em + c m t — ^ //, and c losr t=: ^ + « log - ; and eliminating //, we have the fundamental equation / m\ 0 = Em ^ mty<- — H - c log t + (/ log - J. (260) Differentiating this equation, we obtain / 1 y \ T amt dip =- m\H+ cXo^t-^ «log -J dt ^- dv j.Ie + t Ic + <i - H - c\ogt + a log '-^1 jdm • (261 ) 212 fT. W. Gibhs — Equilibrium of Heterogeneous Substances. whence, by the general equation (88), 1] = m { H+ c log «; + a log — J , (262) am t , . /> = — - (263) c + a - Il—c\ogt-\-alog — \. (264) From (260), by (87) and (91), we obtain 'C, = Em, -\- ')nt\c — H — c log t + a log — ] + p v, and eliminating v by means of (263), we obtain the fundamental equa- tion ? = Eyn + m tic + a - H - {c-^a) log ^ + a log — |. (265) From this, by differentiation and comparison with (92), we may obtain the equations // z=. m (Hi- (c + a) log « — a log — |, (266) a m t ^=-^, (267) lx = E -{- tic + a — H - (e+«) log t + a log — j. (268) The last is also a fundamental equation. It may be written in the form or, if we denote by e the base of the Naperian system of logarithms, E—c—a c + a fi—E p = ae "■ t "" e ""^ (270) The fundamental equation between Xi V-, Pi ^"d m may also be easily obtained ; it is (c+«)log7 =--H+a\og^, (271) ^ ^ * {c-\-a)m m ^ a' ^ ' which can be solved with respect to x- Any one of the fundamental equations (255), (260), (265), (270), and (271), which are entirely equivalent to one another, may be J. W. Gibbs — Eqtcilibriuin of IJeterogeneoKs ySiibstances. 213 regarded as defining an ideal gas. It will be observed that most of these equations might be abbreviated by the use of different con- stants. In (270), for example, a single constant might be used for H—c—a — " C -\- €t a e '^ , and another for ■ ^ The equations have been given in the above form, in order that the relations between the constants occurring in the different equations might be most clearly exhibited. The sum c + a is the specific heat for constant pressure, as appears if we diflerentiate (266) regarding jt> and in as constant.* * We may easily obtain the equation between the temperature and pressure of a saturated vapor, if we know the fundamental equations of the substance both in the gaseous, and in the liquid or solid state. If we suppose that the density and the specific heat at constant pressure of the liquid may be regarded as constant quantities (for such moderate pressures as the liquid experiences while in contact with the vapor), and denote this specific heat by A;, and the volume of a unit of the liquid by V. we shall have for a unit of the liquid t dr/ = k dt, whence 7] = k log t + H\ where H' denotes a constant. Also, from this equation and (97), dfi — - (k log t + R')dt+V dp, whence 11 = kt— kt log t—H't+Vp + E% (a) where E' denotes another constant. This is a fundamental equation for the substance in the liquid state. If (268) represents the fundamental equation for the same sub- stance in the gaseous state, the two equations will both hold true of coexistent liquid and gas. Eliminating u we obtain p H—H' + k—c—a k—c—a, E—E' V p a a a at a t If we neglect the last term, which is evidently equal to the density of the vapor divided by the density of the liquid, we may write C log p=A — Blog t -, A, B, and C denoting constants. If we make similar suppositions in regard to the substance in the solid state, the equation between the pressure and temperature of coexistent solid and gaseous phases wiU of course have the same form. A similar equation will also apply to the phases of an ideal gas which are coexis- tent with two different kinds of solids, one of which can be formed by the combina- tion of the gas with the other, each being of invariable composition and of constant specific heat and density. In this case we may write for one solid /x , -- k't-k't log t- H't + V'p + E', and for the other ji., = k"t-k"t log t-H"t+ V"p + E", and for the gas ^;, = E-^t(c + a-H— (c + a) log f + a log — j. 214 J. W. Gibbs — Equilibriiini of Heterogeneous Substances. The preceding fundamental equations all apply to gases of constant composition, for which the matter is entirely determined by a single Now if a unit of the gas unites with the quantity /i of the first solid to form the quantity \ + l of the second it will be necessary for equilibrium (see pages 121, 122) that ^3+A^, =(1 +A)//.,. Substituting the values of /i,, //._,, //;j, given above, we obtain after arranging the terms and dividing by at when loo; — = A- Bios t + D — , ^ a ^ t I ' _ H+AH'-{l+l)H"-c-a-lk' + {l + 'A)k' „ (WA)Jc"-lk'-c~a a E+lE'-{\+l)E" (1+A)F"-AF' 0 — , L) — ■ . a a We may conclude from this that an equation of the same form may be applied to an ideal gas in equilibrium with a liquid of wliich it forms an independently variable component, wlien the specific heat and density of the liquid are entirely determined by its composition, except that the letters A. B, C, and D must in this case be under- stood to denote quantities which vary with the composition of the liquid. But to consider the case more in detail, we have for the liquid by (a) - - —u=ki-kt loo; t-H't+ Vp + E', m where k, H% V, E' denote quantities which depend only upon the composition of the liquid. Hence, we may write C = bt-kt log (~m + V]) + E, where k, H, V, and B denote functions of m^, rwj, etc. (the quantities of the several components of the liquid). Hence, by (92), dk ^ dk , dH. dV dE // , = ^T—t- -—t log If — -- — 1+ -^—2}+ 1— . am I dm, dm-i dm, dm, If the component to which this potential relates is that which also forms the gas, we shall have by (269) •p H—c—a c + a, /"j— -^ log ^ = + log t+'-^-- . a a a at Eliminating /^ i , we obtain the equation log^=^-51ogi-y+ i>-y-, in which A, B, C, and D denote quantities which depend only upon the composition of the liquid, viz : \ I d-H. rfk A = —[ H- c-a-\-^,— a \ dm, dm,. B=L(^-c-a) a \dm, )' J. W. Gibhs — Equilihi'mDi of Heterogeneous Substances. 215 variable (m). We may obtain correspoiulincj fundamental equations for a mixture of gases, in which the proportion of the components shall be variable, from the following considerations. It is a rule which admits of a very general and in many cases very exact experimental verification, that if several liquid or solid sub- stances which yield difi^erent gases or vapors are simultaneously in equilibrium with a mixture of these gases (cases of chemical action between the gases being excluded,) the pressure in the gas-mixture is equal to the sum of the pressures of the gases yielded at the same temperature by the various liquid or solid substances taken separately. Now the potential in any of the liquids or solids for the substance which it yields in the form of gas has very nearly the same value when the liquid or solid is in equilibrum with the gas-mixture as when it is in equilibrium with its own gas alone. The difference of the pressure in the two cases will cause a certain difference in the values of the potential, but that this difference will be small, we may infer from the equation C^) =(,*) , (272) \ dp ft, m \dm^lt,p,m ^ which may be derived from equation (92). In most cases, there will be a certain absorption by each liquid of the gases yielded by the others, but as it is well known that the above rule does not apply to cases in which such absorption takes place to any great extent, we may conclude that the effect of this circumstance in the cases with which we have to do is of secondary importance. If we neglect the slight differences in the values of the potentials due to these cii-cum- stances, the rule may be expressed as follows : The pressure in a mixture of different gases is equal to the sum of the pressures of the different gases as existing each by itself at the same temperature avid with the same value of its potential. To form a precise idea of the practical significance of the law as thus stated with reference to the equilibrium of two liquids with a mixture of the gases which they emit, when neither liquid absorbs the gas emitted by the other, we may imagine a long tube closed at each end and bent in the form of a W to contain in each of the descending C= - (^-y-l ^=- ^ • a \ dm I / a am , With respect to some of the equations which have here been deduced, the reader may compare Professor Kirchhoff " Ueber die Spannung des Dampfes von Mischungen aus Wasser und Schwefelsaure," Pogg. Ann., vol. civ. (18.58), p. 612 ; and Dr. Raukine "On Saturated Vapors,'' Phil. Mag., vol. xxxi. (1866), p. 199. 216 J. W. Gibbs — Equilibrium of Heterogeneous Substances. loops one of the liquids, and above these liquids the gases which they emit, viz., the separate gases at the ends of the tube, and the mixed gases in the middle. We may suppose the whole to be in equilibrium, the difference of the pressures of the gases being balanced by the proper heights of the liquid columns. Now it is evident from the principles established on pages 203-210 that the potential for either gas will have the same value in the mixed and in the separate gas at the same level, and therefore according to the rule in the form which we have given, the pressure in the gas-mixture is equal to the sum of the pi'essures in the separate gases, a/^ (^Aese joressiwes being measured at the same level. Now the experiments by which the rule has been established relate rather to the gases in the vicinity of the surfaces of the liquids. Yet, although the differences of level in these surfaces may be considerable, the corresponding differences of pres- sure in the columns of gas will certainly be very small in all cases which can be I'egarded as falling under the laws of ideal gases, for which very great pressures are not admitted. If we apply the above law to a mixture of ideal gases and distin- guish by subscript numerals the quantities relating to the different gases, and denote by ^'^ the sum of all similar terms obtained by changing the subscript numerals, we shall have by (270) -ff, — Cj— a, Ci+a, fi^—E, (a, a, ttit \ , a^ e t e /, (273) It will be legitimate to assume this equation provisionally as the fundamental equation defining an ideal gas-mixture, and afterwards to justify the suitableness of such a definition by the properties which may be deduced from it. In particular, it will be necessary to show that an ideal gas-mixture as thus defined, when the proportion of its components remains constant, has all the properties which have already been assumed for an ideal gas of invariable composition ; it will also be desirable to consider more rigorously and more in detail the equilibrium of such a gas-mixture with solids and liquids, with respect to the above rule. By differentiation and comparison with (98) we obtain = ^^\ («,+«,- ^-7—^) e t e ;, (274) V t J. W. Gihhs — Eqidllhriiua of Iltterogeneous /Substances. 2 I 7 H-i—Cj—a., c^ l^i—E.2 )■ etc. (275) Equations (275) indicate that the relation between the temperature, the density of any compcfnent, and the potential for that component, is not aifected by the presence of the other components. They may also be written etc. ) Eliminating yu,, /<2, etc. from (273) and (274) by means of (275) and (276), we obtain (277) 7= ^lyn^ir^ +M,c,log «+m,«ilog ^j. (278) E({uation (277) expresses the familiar principle that the pressure in a gas-mixture is equal to the sum of the pressures which the component gases would possess if existing separately with the same volume at the same temperature. Equation (278) expresses a similar principle in regard to the entropy of the gas-mixture. From (276) and (277) we may easily obtain the fundamental equa- tion between //', t, v, m^^ ni^, etc. For by substituting in (94) the values of jo, ji^, /.i^, etc. taken from these equations, we obtain 'p=2^(^£.\m,-\-m^t | c, -^, -c.log « + «,log "^j). (279) If we regard the proportion of the various components as constant, this equation may be simplified by writing m for ^j wZj, c m for ^ J (c , m J ), am for 2^ (a^m^), Em for ^j (£', mj), and Hm-am log m for ^j (H^ m^—a^ m^ log rn^). The values of c, a, -E, and JT, will then be constant and m will denote the total quantity of gas. As the equation Avill thus be reduced to the Trans. Conn. Acad., Vol. III. 28 April, 1876. 218 (./. W. (jribhs — Equilibrkiiu of Heterogeneous Suhstances. form of (260), it is evident that an ideal gas-mixture, as defined by (278) or (279), when the proportion of its components remains un- changed, will have all the properties which we have assumed for an ideal o-as of invariable composition. The relations between the specific heats of the gas-mixture at constant volume and at constant pressure and the specific heats of its components are expressed by the equations c = ^'^-\ (280) m and ,;^_«=^' !!hj£i+^. (281) We have already seen that the values of t, v, m^, /.i^ in a gas- mixture are such as are possible for the component G^ (to which /«j and /<j relate) existing separately. If we denote by jOj, j/^, //'j, £,, ^'j, Cj the connected values of the several quantities which the letters indicate determined for the gas 6^j as thus existing sepa- rately, and extend this notation to the other components, we shall have by (273), (274), and (279) whence by (87), (89), and (91) The quantities p, //, '/', €, j, ? relating to the gas-mixture may therefore be regarded as consisting of parts which may be attrib- uted to the several components in such a manner that between the parts of these quantities which are assigned to any component, the quantity of that component, the potential for that component, the temperature, and the volume, the same relations shall subsist as if that component existed separately. It is in this sense that we should understand the law of Dalton, that every gas is as a vacuum to every other gas. It is to be remarked that these relations are consistent and pos- sible for a mixture of gases which are not ideal gases, and indeed without any limitation in regard to the thei'modynamic properties of the individual gases. They are all consequences of the law that the pressure in a mixtuz-e of dilFerent gases is equal to the sum of the pressures of the different gases as existing each by itself at the same tempei'ature and with the same value of its potential. For let Pii V\i ^n '/'i' /I'l' ^1 ' P2-> etc.; etc. be defined as relating to the different gases existing each by itself with the same volume, tem- perature, and potential as in the gas-mixture ; if '/=-i'/n t=^^'/\^, (282) ") x=^ai^ '^ = ^\'ii- (283) ./ W. (jTibhs — EquUlbrlwn of Heteroyeneoas ISuhstaHceti. 219 the,, l^\ =iP): and therefore, by (98), the quantity of any component gas <t ^ in tlie gas-mixture, and in the separate gas to which p^, //j, eic. relate, is the same and may be denoted by the same symbol )ii ^. Also whence also, by (93)-(96), All the same relations will also hold true whenever the value of t/^ for the gas-mixture is equal to the sum of the values of this func- tion for the several component gases existing each by itself in the same quantity as in the gas-mixture and with the temperature and volume of the gas-mixture. For if ^^j, //j, fj, i/\, Xi-> ^i ? 2^21 etc. ; etc. are defined as relating to the components existing thus by themselves, we shall have 'I- =^ lip I, whence \drn^ /i, V, m \dm^ ft, v Therefore, by (88), the potential //j has the same value in the gas- mixture and in the gas G^ existing separately as supposed. More- over, '^=^ idiJv, nT ~ ^A~df)v, ra " ^'^'^^' whence Whenever different bodies are combined without communication of work or heat between them and external bodies, the energy of the body formed by the combination is necessarily equal to the sum of the energies of the bodies combined. In the case of ideal gas-mixtures, when the initial temperatures of the gas-masses which are combined * A subscript m after a differential coefficient relating to a body having several independently variable components is used here and elsewhere in this paper to indi- cate that each of the quantities m^,m2, etc., unless its differential occurs in the expression to which the suffix is applied, is to be regarded as constant in the differ- entiation. 220 J. W. Gibbs — Equilibrium, of Heterogeneous Substances. are the same, (whether these gas-masses are entirely different gases, or gas-mixtures differing only in the proportion of their components,) the condition just mentioned can only be satisfied when the tempera- ture of the resultant gas-mixture is also the same. In such com- binations, therefore, the final temperature will be the same as the initial. If we consider a vertical column of an ideal gas-mixture which is in equilibrium, and denote the densities of one of its components at two different points by y^ and ;//, we shall have by (275) and (234) i^i-/"/ g[h'-h) ^=e "'* =e ""'* . (284) From this equation, in which we may regard the quantities distin- guished by accents as constant, it appears that the relation between the density of any one of the components and the height is not affected by the presence of the other components. The work obtained or expended in any reversible process of com- bination or separation of ideal gas-mixtures at constant temperature, or when the temperatures of the initial and final gas-masses and of the only external source of heat or cold which is used are all the same, will be found by taking the difference of the sums of the values of ip for the initial, and for the final gas-raasses. (See pages 145, 146). It is evident from the form of equation (279) that this work is equal to the sum of the quantities of work which would be obtained or expended in producing in each different component existing separately the same changes of density which that component experiences in the actual process for which the w^ork is sought.* We will now return to the consideration of the equilibrium of a liquid with the gas which it emits as affected by the presence of difterent gases, when the gaseous mass in contact with the liquid may be regarded as an ideal gas-mixture. It may first be observed, that the density of the gas which is emitted by the liquid will not be affected by the presence of other gases which are not absorbed by the liquid, when the liquid is pro- tected in any way from the pressure due to these additional gases. This may be accomplished by separating the liquid and gaseous * This result has been given by Lord Rayleigh, (Phil. Mag., vol. xlix, 1875, p. 311). It will be observed that equation (279) might be deduced immediately from this principle in connection with equation (260) which expresses the properties ordinarily assumed for perfect gases. ./ W. Gibbs — Equilibrmm of Heterogeneous Substances. 221 masses by a diaphragm which is permeable to the liquid. It will tlien be easy to maintain the liquid at any constant pressure which is not greater than that in the gas. The potential in the liquid for the substance which it yields as gas will then remain constant, and there- fore the potential for the same substance in the gas and the density of this substance in the gas and the part of the gaseous pressure due to it will not be affected by the other components of the gas. But when the gas and liquid meet under ordinary circumstances, i. e., in a free plane s\irfi\ce, the pressure in both is necessarily the same, as also the value of the potential for any common component aSj. Let us suppose the density of an insoluble component of the gas to vary, while the composition of the liquid and the temperature remain unchanged. If we denote the increments of pressure and of the potential for S^ by djj and c?/<j, we shall have by (272) \dp lt,m \dmjt,p,m the index (l) denoting that the expressions to which it is affixed refer to the liquid. (Expressions without such an index will refer to the gas alone or to the gas and liquid in common.) Again, since the gas is an ideal gas-mixture, the relation between p^ and /u^ is the same as if the component aS'j existed by itself at the same temperature, and therefore by (268) (/// J = a J t d log p^. Therefore (dv \^^^ - — I dp. (285) dmjt,p,m ^ This may be integrated at once if we regard the differential coeffi- cient in the second member as constant, which will be a very close approximation. We may obtain a result more simple, but not quite so accurate, if we write the equation in the form -^ dp, (286) dm ^ ft, 2), m where ;/j denotes the density of the component /S^ in the gas, and integrate regarding this quantity also as constant. This wdll give where jt?/ and p/ denote the values of ^j and p -when the insoluble component of the gas is entirely wanting. It will be observed that p—p' is nearly equal to the pressure of the insoluble component, in the phase of the gas-mixture to which pi relates. /S', is not neces- 222 J. W. Gibbs — Equilibrmm of Heterogeneous Substances. sarily the cmly common component of the gas and liquid. If there are others, we may find the increase of the part of the pressure in the gas-mixture belonging to any one of them by equations differing from the last only in the subscript numerals. Let us next consider the effect of a gas which is absorbed to some extent, and which must therefore in strictness be regarded as a com- ponent of the liquid. We may commence by considering in general the equilibrium of a gas-mixture of two components /8, and IS^ with a liquid formed of the same components. Using a notation like the previous, we shall have by (98) for constant temperature, and dp = y^^^ di.i^-\-yf^ dji^ ; whence Now if the gas is an ideal gas-mixture, a.t -, dp. -, , a^t ^ dp„ du.=i —^ dpx-=. -^-', and au^ =i — ^ rt», = -=-^, ^' Pv Vx Pz 72 therefore l^-\]^dp,= [\-^^l dp^. (288) We may now suppose that S^ is the principal component of the liquid, and aS's is a gas which is absorbed in the liquid to a slight extent. In such cases it is well known that the ratio of the densities of the substance S2 in the liquid and in the gas is for a given tem- perature approximately constant. If we denote this constant by A, we shall have ^r.L- ^^^dp^={\-A)dp^. (289) It would be easy to integrate this equation regarding ;/ j as variable, but as the variation in the value of », is necessarily very small we (L) shall obtain sufficient accuracy if we regard }^i as well as ;(/i as con- stant. We shall thus obtain (^'^-l)(/>,-^p,')=(l-^)i5„ (290) where ^1' denotes the pressure of the saturated vapor of the pure liquid consisting of S^. It will be observed that when ^=1, the presence of the gas S^ will not affect the pressure or density of the gas S^. When ^<^1, the pressure and density of the gas S^ are greater than if S.^ were absent, and when A^\, the revei-se is true. ./. W. Gibbs — Equilibrium of Heterogeneous Substances. 223 The properties of an ideal gas-mixture (according to the definition which we have assumed) when in equilibrium with liquids or solids have been developed at length, because it is only in respect to these properties that there is any variation from the properties usually attributed to perfect gases. As the pressure of a gas saturated with vaporis usually given as a little less than the sum of the pressure of the gas calculated from its density and that of saturated vapor in a space otherwise empty, while oxir formulae would make it a little more, when the gas is insoluble, it would appear that in this respect our formulae are less accurate than the rule which would make the pressure of the gas saturated with vapor equal to the sum of the two pressures mentioned. Yet the reader will observe that the magnitude of the quantities concerned is not such that any stress can be laid upon this circumstance. It will also be observed that the statement of Dalton's law which we have adopted, while it serves to complete the theory of gas-mixtui-es (with respect to a certain class of properties), asserts nothing with reference to any solid or liquid bodies. But the common rule that the density of a gas necessary for equilibrium with a solid or liquid is not altered by the presence of a different gas which is not absorbed by the solid or liquid, if construed strictly., will involve consequences in regard to solids and liquids which are entirely inadmissible. To show this, we will assume the correctness of the rule mentioned. Let aS'j denote the common component of the gaseous and liquid or solid masses, and /Sg the insoluble gas, and let quantities relating to the gaseous mass be distinguished when necessary by the index (g), and those relating to the liquid or solid by the index (l). Now while the gas is in equilibrium with the liquid or solid, let the quantity which it contains of ^'2 receive the increment dm^., its volume and the quantity which it contains of the other component, as well as the temperature, remaining constant. The potential for S^ in the gaseous mass will receive the increment \ani2l t, V, m and the pressure will receive the increment \dm. I dnic [dm 2 ft, V, m Now the liquid or solid remaining in equihbrium with the gas must experience the same variations in the values of /u j and p. But by (272) \ dpJt,m \dmjt,j>, 224 J. W. Gibbs — Equilibrium of Heterogeneous ISubstances. Therefore, \dmjt,2], m i dp Y*^^ \dm2/t, V, 1 It will be observed that the first member of this equation relates solely to the liquid or solid, and the second member solely to the gas. Now we may suppose the same gaseous mass to be capable of equilibrium with several diiferent liquids or solids, and the first mem- ber of this equation must therefore have the same value for all such liquids or solids ; which is quite inadmissible. In the simplest case, in which the liquid or solid is identical in substance with the vapor which it yields, it is evident that the expression in question denotes the reciprocal of the density of the solid or liquid. Hence, when a gas is in equilibrium with one of its components both in the solid and liquid states (as when a moist gas is in equilibrium with ice and water), it would be necessary that the solid and liquid should have the same density. The foregoing considerations appear sufiicient to justify the defini- tion of an ideal gas-mixture which we have chosen. It is of course immaterial whether we regard the definition as expressed by equation (273), or by (279), or by any other fundamental equation which can be derived from these. The fundamental equations for an ideal gas-mixture corresponding to (255), (265), and (271) may easily be derived from these equations by using inversely the substitutions given on page 217. They are ^,(c. m,) log '-^^£^=r,-\.2, {a,m, log^-^,^J, (291) = V^^.{-.>n^^og^-^^-B,m,Y (292) - 2^ {c,m,+a, m,) t log t +^\ [ct, m, t log ^h^^^^). (293) The components to which the fundamental equations (273), (279), (291) (292), 293) refer, may themselves be gas-mixtures. We may for example apply the fundamental equations of a binary gas-mixture J. W. Gibbs — Eqidlihriut)i of Heterogeneous Sultstuitces. 225 to a mixture of hydrogen and air, or to any ternary gas-mixture in wliich the proportion of two of the components is fixed. In fact, tlie form of equation (279) which applies to a gas-mixture of any pai'ticu- lar number of components may easily be reduced, when the propor tions of some of these components are fixed, to the form whicli ai)plies to a gas-mixture of a smaller niunber of components. The necessary substitutions will be analogous to those given on page 217. But the components must be entirely different from one another with respect to the gases of which they are formed by mixture. We cannot, for example, apply equation (279) to a gas-mixture in which the com- ponents are oxygen and air. It would indeed be easy to form a fundamental equation for such a gas-mixture with reference to the designated gases as components. Such an equation might be derived from (279) by the proper substitutions. But the result would be an equation of more complexity than (279). A chenncal compound, however, with respect to Dalton's law, and with respect to all the equations which have been given, is to be regarded as entirely differ- ent from its components. Thus, a mixture of hydrogen, oxygen, and vapor of water is to be regarded as a ternary gas-mixture, having the three components mentioned. This is certainly true when the quanti- ties of the compound gas and of its components are all independently variable in the gas-mixture, without change of temperature or pres- sure. Cases in which these quantities are not thus independently variable will be considered hereafter. Inferences in regard to Potentials iti Liquids and Solids. Such equations as (264), (268), (276), by which the values of potentials in pure or inixed gases may be derived from quantities capable of direct measurement, have an interest which is not confined to the theory of gases. For as the potentials of the independently variable components which are common to coexistent liquid and gas- eous masses have the same values in each, these expressions will generally afford the means of determining for liquids, at least ap- proximately, the potential for any independently variable compon- ent Avhich is capable of existing in the gaseous state. For although every state of a liquid is not such as can exist in contact with a gaseous mass, it will always be possil)le, when any of the components of the liquid are volatile, to bring it by a change of pressure alone, its temperature and composition remaining unchanged, to a state for which there is a coexistent phase of vapor, in which Trans. Conn. Acad., Vol. III. 29 May, KSTfi. 226 ./. W. Glbbs — EqulUbrium of Ileteroyeiieoits Substances. the values of the potentials of the volatile components of the liquid may be estimated from the density of these substances in the vapor. The variations of the potentials in the liquid due to the change of pressure will in general be quite trifling as compared with the variations which are connected with changes of temperature or ot composition, and may moreover be readily estimated by means of equation (272). The same considerations will apply to volatile solids with respect to the determination of the potential for the substance of the solid. As an application of this method of determining the potentials in liquids, let us make use of the law of Henry in regard to the absorption of s^ases by liquids to determine the relation between the quantity of the gas contained in any liquid mass audits potential. Let us consider the liquid as in equilibrium with the gas, and let m'-^^ denote the quantity of the gas existing as such, rn'-^^ the quantity of the same substance contained in the liquid mass, yUj the potential for this substance common to the gas and liquid, v^^'> and v^^^ the volumes of the gas and liquid. When the absorbed gas forms but a very small part of the liquid mass, we have by Henry's law ^ = .4 J-, (204) where ^1 is a function of the temperature ; and by (-'76) ni (G) ;., = i?+CMog-^^, (295) v^ B and C also denoting functions of the tenq>erature. Therefore m (L) It will be seen (if we disregai-d the difference of notation) that this equation is equivalent in form to (216), which was deduced from a jorior* considerations as a probable relation between the quantity and the potential of a small component. When a liquid absoi'bs several gases at once, there will be sevei'al equations of the form of (296), which will hold true simultaneously, and which we may regard as equivalent to equations (217), (218). The quantities A and C in (216), with the corresponding quantities in (21 7), (218), were regarded as functions of the temperature and pressure, but since the potentials in liquids are but little affected by the pressure, we might anticipate that these quantities in the case of liquids miglit be regarded as func- tions of the temperature alone. J. W. Gihbs — EquUihrhDii of Ileteroqeneoun. Substances. 227 In regard to equations (216), (2lV), (218), we may now observe that by (264) and (276) they are shown to hokl true in ideal gases or gas-mixtures, not only for components which form only a small part of the whole gas-mixture, but without any such limitation, and not only approximately but absolutely. It is noticeable that in this case quantities A and C are functions of the temperature alone, and do not even depend upon the nature of the gaseous mass, except upon the particular component to which they relate. As all gaseous bodies are generally supposed to approximate to the laws of ideal gases when sufficiently rarefied, we may regard these equations as approximately valid for gaseous bodies in general when the density is sufficiently small. When the density of the gaseous mass is very great, but the separate density of the comjionent in question is small, the equa- tions will probably hold true, but the values of A and C may not be entirely independent of the pressure, or of the composition of the mass in respect to its principal components. These equations will also apply, as we have just seen, to the potentials in liquid bodies for com- ponents of which the density iu the liquid is very small, whenever these components exist also in the gaseous state, and conform to the law of Henry. This seems to indicate that the law expressed by these equations has a very general application. Considerations relating to the Increase of Entropy due to the Mixture of Gases by Diffusion. From equation (278) we may easily calculate the increase of entropy which takes place when two different gases are mixed by diffusion, at a constant temperature and pressure. Let us suppose that the quantities of the gases are such that each occupies initially one half of the total volume. If we denote this volume by F, the increase of entropy will be V V m , a^ log F-f mg a^ log F- m^ a^ log — - m^ a^ log -^, or {7n^ «, + >«2 ^'2) log 2. p F -, pV Now m^a-^ = ---, and m.^ a^ = — y. Therefore the increase of entropy may be represented by the expres- sion ^- log 2. (297) It is noticeable that the value of this expression does not depend upon the kinds of gas which are concerned, if the quantities are such as has been supposed, except that the gases which are mixed must be 228 ./ W. Gihhs — Equilibrhim of Heterogeneous Substances. of different kinds. If we should bring into contact two masses of the same kind of gas, they would also mix, but there would be no in- crease of entropy. But in regard to the relation which this case bears to the preceding, we must bear in mind the following considera- tions. When we say that when two different gases mix by diffusion, as we have supposed, the energy of the whole remains constant, and the entropy receives a certain increase, we mean that the gases could be separated and brought to the same volume and temperature which they had at first by means of certain changes in external bodies, for example, by the passage of a certain amount of heat from a warmer to a colder body. But when we say that when two gas-masses of the same kind are mixed under similar circumstances there is no change of energy or entropy, we do not mean that the gases which have been mixed can be separated without change to external bodies. On the contrary, the separation of the gases is entirely impossible. We call the energy and entropy of the gas-masses when mixed the same as when they were unmixed, because we do not recognize any difference in the si\bstance of the two masses. So when gases of different kinds are mixed, if we ask what changes in external bodies are necessary to bring the system to its original state, we do not mean a state in which each particle shall occupy more or less exactly the same posi- tion as at some previous epoch, but only a state which shall be undistinguishable from the previous one in its sensible properties. It is to states of systems thus incompletely defined tliat the problems of thermodynamics relate. But if such considerations explain why the mixture of gas-masses of the same kind stands on a different footing from the mixture of gas-masses of different kinds, the fact is not less significant that the increase of entropy due to the mixture of gases of different kinds, in such a case as we have supposed, is indej^endent of the nature of the gases. Now we may without violence to the general laws of gases which are embodied in our equations suppose other gases to exist than such as actually do exist, and there does not appear to be any limit to the resemblance which there might be between two such kinds of gas. But the increase of entropy due to the mixing of given volumes of the gases at a given temperature and pressure would be independent of the degree of similarity or dissimilarity between them. We might also imaoine the case of two gases which should be absolutely identi- cal in all the properties (sensible and molecular) which come into play while they exist as gases either pure or mixed with each other, J. ^V. Glbbs — Equllihriioti of Ileievogeneous SuhsUtvces. 229 but which sliouhl differ in respect to the attractions between tlieir atoms antl the atoms of some other substances, and therefore in tlieir tendency to combine Mith such sul)stances. In tlie mixture of such gases by diffusion an increase of entropy wouhl take ))hice, although the process of mixture, dynamically considered, might be absolutely identical in its minutest details (e\ en with i-espect to the i)recise path of each atom) with processes which might take ]>lace without any increase of entropy. In sucli respects, entropy stands strongly con- trasted with energy. Again, when such gases have been mixed, there is no more irapossil)ility of the separation of the two kinds of molecules in virtue of their ordinary motions in the gaseous mass without any especial external influence, than there is of the separation of a lumio- geneous gas into the same two parts into which it has once been divided, after tliese have once been mixed. In other words, the impossibility of an uncompensated decrease of entropy seems to be reduced to improbability. There is perhaps no fact in tlie molecular theory of gases so well established as that the number of molecules in a given volume at a given temperature and .pressure is the same for every kind of gas when in a state to which the laws of ideal gases apply. Hence the quantity — — in (297) must be entirely determined by the number of molecules which are mixed. And the increase of entropy is therefore determined by the number of these molecules and is independent of their dynamical condition and of the degree of difference between them. The result is of the same nature when the volumes of the gases which are mixed are not equal, and when more than two kinds of gas are mixed. If we denote by v^, v^? etc., the initial volumes of the different kinds of gas, and by V as before the total volume, the increase of entropy may be written in the form :E^ {m^ a^) log V- :S^ {m, a, logy J. And if we denote by r,, rg, etc., the numbers of the molecules of the several different kinds of gas, we shall have r^ = (Jm^ «!, ?'2 = Cm^ a.^, etc., where (J denotes a constant. Hence V ^:V:: m^a^ : 2 ^{m^a.^) ::1\ : ^, r^ ; and the increase of entropy may be written ^^^ilog^i^i -^i(^ilog^i)^ (298) C 230 J. W. Gibbs — Equilibrhim of Heterogeneous ^Substances. The Phases of Dissipated Energy of an Ideal Gas-ndxtare loith Components v^hivh are Vhemically Related. We will now pass to the considevation of the phases of dissipated energy (see page 200) of an ideal gas-mixture, in which the number of the proximate components exceeds that of the ultimate. Let us first suppose that an ideal gas-mixture has for proximate components the gases 6r,, 6^3, and 6^g, the units of which are denoted by @^, @2, @3, and that in ultimate analysis @3 = A,®, +A2@2, (299) A, and A2 denoting positive constants, such that Aj + Ag = 1. The phases which we are to consider are those for uiiich the energy of the gas-mixture is a minimum for constant entropy and volume and constant quantities of G^ and 6r g, as determined in ultimate analysis. For such phases, by (86), /^i 8m ^ 4- //g 6m. ^ + fx^ Sm^^O. (300) for such values of the variations as do not affect the quajitities of (tj and 6rg as determined in ultimate analysis. Values of dm^, 6ino, (Si)ip^ proportional to A,, A,, — 1, and only such, are evidently consistent with this restriction : therefore Aj Ml + Ao ^2 = 1^2- (301) If we substitute in this equation values of fi^, /^2? /'a taken from (2*76), we obtain, after arranging the terms and dividing by t, ^1 «i log V+ '^^ ""■' ^""^ "V ~ "'^ ^""^ "zT ^ -^+ Slog^ — ?, (302) where ^ = A, JTj + Aa^o — ^3-A,Cj-A2C2-f Cg-Ajffj-Aatta + ^s^ (•'^03) ^czrAjCj-fA^eo-Cg, (304) G~\^E^-\-\„E^— E^. (305) If we denote by /^, and fi^ the volumes (determined under stand- ard conditions of temperature and pressure) of the quantities of the gases G ^^ and G^ which are contained in a unit of volume of the gas 6^3, we shall have /A = ^'\and /i. = ^-|^, (300) and (302) will reduce to the form log "^^ T\ a ^ = - + — log ^ - — . (307) ^\nj' A B. 'C m„ V 3 "3 "s J. W. Glbbs — J^fjiulibriuiti of Ileteroyeneous ISubstaiices. 231 Moreover, as by (277) pv = {(fi hi ^ + a.^ »*3 + rtg iii^) f, (;508) we have on eliminating v /:?, ii.2 fi, + /3.^ — 1 loff ^1 '"2 P . »i3 («! m, -{- «3 Wig -f «3 ?>?.3) ^ ^' C 3 3 ^^ 3 ^ where ^' == A J Cj -f- ^ 2 <^2 - *'3 + '^ 1 ^' I + '^ 2 ^i! — «3- ("^ 1 ") It will be observed that the quantities /ij, /J.^ will always be posi- tive and have a simple relation to unity, and that the value of /i, -f /!^2 ~ 1 will be positive or zero, according as gas G^ is formed of (tj and G2 with or without condensation. If we should assume, according to the rule often given for the specific heat of compound gases, that the thermal capacity at constant volume of any quantity of the gas 6^3 is equal to the sum of the thermal capacities of the quantities which it contains of the gases G^ and G.^, the value of B would be zero. The heat evolved in the formation of a miit of the gas 6^3 out of the gases G ^ and G2, without mechanical action, is by (283) and (257) A , (c, « + ^1) + A2 (c'2 t-^KJ - (C3 t+U^), or Bt -\- a, which will reduce to C when the above relation in regard to the specific heats is satisfied. In any case the quantity of heat thus evolved divided by ^3 t^ will be equal to the differential coefficient of the second member of equation (307) with respect to t. Moreover, the heat evolved in the formation of a unit of the gas G.^ out of the gases 6r, and G2 imder constant pressure is Bt + C+A,«i t 4- A..a2t-a,,t=zB't-\-C, which is equal to the differential coefficient of the second member of (309) with respect to t, multiplied by a^ t'^ . It appears by (307) that, except in the case when ji ^ + f-j^ = 1, for any given finite values oi' iii ^, in.,, ni^, and t (infinitesimal values being excluded as well as infinite), it will always be possible to assign such a finite value to v that the mixture shall be in a state of dissipated energy. Thus, if Ave regard a mixture of hydrogen, oxy- gen, and vapor of water as an ideal gas-mixture, for a mixture con- taining any given quantities of these three gases at any given tem- 232 J. W. Gibbs — Equilibrium of Heterogeneous Substances. perature there will be a certain volume at which the mixture will be in a state of dissipated energy. In such a state no such phenomenon as explosion will be possible, and no formation of water by the action of platinum. (If the mass should be expanded beyond this volume, the only possible action of a catalytic agent would be to resolve the water into its components.) It may indeed be true that at ordinary temperatures, except when the quantity either of hydrogen or of oxygen is very small compared with the quantity of water, the state of dissipated energy is one of such extreme rarefaction as to lie entirely beyond our power of experimental verification. It is also to be noticed that a state of great rarefaction is so unfavorable to any condensation of the gases, that it is quite probable that the catalytic action of platinum may cease entirely at a degree of rarefaction far short of what is necessary for a state of dissipated energy. But with respect to the theoretical demonstration, such states of great rarefac- tion are precisely those to which we should suppose that the laws of ideal gas-mixtures would apply most perfectly. But when the compound gas G^ is formed of 6r, and G^ without condensation, (i. e., when /i, -\- (i.^ =r 1,) it appears from equation (307) that the relation between iit^, m.^, and rn^ which is necessary for a phase of dissipated energy is determined by the temperature alone. In any case, if we regard the total quantities of the gases G^ and 6^2 (^s determined by the ultimate analysis of the gas-mixture), and also the volume, as constant, the quantities of these gases which appear uncombined in a phase of dissipated energy will increase with the temperature, if the formation of the compound 6^3 without change of volume is attended with evolution of heat. Also, if we regard the total quantities of the gases G^ and G^, and also the pressure, as constant, the quantities of these gases which appear un- combined in a phase of dissipated energy, will increase with the temperature, if the formation of the compound G^ under constant pressure is attended with evolution of heat. If J5 = 0, (a case, as has been seen, of especial importance), the heat obtained by the formation of a unit of G^ out of G^ and G2 without change of volume or of temperature will be equal to C. If this quantity is positive, and the total quantities of the gases G^ and G2 and also the volume have given finite values, for an infinitesimal value of t we shall have (for a phase of dissipated energy) an infinitesimal value either pf m^ or of ^2, and for an infinite value of t we shall have finite (neither in- finitesimal nor infinite) values of m,, m^, and m^. But if we suppose the pressure instead of the volume to have a given finite value (with J. W. (ribbs — Eqiiilibriiim of Heterogeneous ^<ubstances. 2;j3 suppositions otherwise tlie same), we shall have for infinitesimal values of ^ an infinitesimal value either of w/ , or Wig, and for infinite values of t finite or infinitesimal values of rit^ according as /j, -|- /i^ is equal to or greater than unity. The case which we have considered is that of a ternary gas-mix- ture, but our results may easily be generalized in this respect. In fact, whatever the number of component gases in a gas-mixture, if there are relations of equivalence in ultimate analysis l)etween these components, such relatioTis may be expressed by one or more equa- tions of the form A , (S^ , + A2 C'^io + A 3 (SV^ + etc. in 0, (31 1) where @j, (S^g? ^^c. denote the units of the various component gases, and A,, A 2, etc. denote positive or negative constants such that 2^ A , =: 0. From (311) with (R6) we may derive for phases of dis- sipated energy, Aj //j + A2 11-2 + ^^3 /'3 + ^tc. =0, or ^^j (A,//,) = 0. (312) Hence, by (276), ^,[^,a,\og~^) = A + B\o^t ^ J, (313) where A, B and (' are constants determined by the equations A = 2, {X,II, - A,c, - X,a,), (314) B=^,{\,c,\ (315) C=2,{X,E,). (316) Also, since 2^ ^^ — ^1 ('"'^ 1 "' 1 ) ^ ^' J (A , «j log «? , ) — ^ (A ^ a , ) log JS" J (« J m , ) -f ^^(A,«,)log/> = .4 + ^'log<--^, (317) where B' = 2, (A,c,-f A,«J. (318) If there is more than one equation of the form (311), we shall have more than one of each of the forms (313) and (317), which will hold true simultaneously for phases of dissipated energy. It will be observed that the relations necessary for a phase of dis- sipated energy between the volume and temperature of an ideal gas- mixture, and the quantities of the components which take part in the chemical processes, and the pressure due to these components, are not affected by the presence of neutral gases in the gas-mixture. Trans. Conn. Acad., Vol. III. ::0 May, 187(3. '234 J. W. Gibbs — Equilibrium of Heterogeneous Substances. From equations (312) and (234) it follows that if there is a phase of dissipated energy at any point in an ideal gas-mixture in equili- In-iuni under the iutiuence of gravity, the whole gas-mixture must consist of such phases. The equations of the phases of dissipated energy of a binary gas- mixture, the components of which are identical in substance, are com- paratively simple in form. In this case the two components have the same potential, and if we write /i for — (the ratio of the volumes of equal quantities of the two components under the same conditions of temperature and pressure), we shall have log ^ = -^- H log t - — -, (319) 3—1 a, cto «2 ^ m^ V d ^ <£ log i-^^i — — 1 lost- — : ; (-3^^) / , V /3— 1 «2 «P «2 ^ where ^ = c, -C2, i?' = Ci — Cg +«j — rtg, (322) C=iE^~E^. (323) Gas-mixtures with Convertible Gom,ponents. The equations of the phases of dissipated energy of ideal gas-mix- tures which have components of which some are identical in ultimate analysis to others have an especial interest in relation to the theory of gas-mixtures in which the components are not only thus equivalent, but are actually transformed into each other within the gas-mixture on variations of temperature and pressure, so that quantities of these (proximate) components are entirely determined, at least in any per- manent phase of the gas-mixture, by the quantities of a smaller number of ultimate components, with the temperature and pi-essure. Such gas-inixtures may be distinguished as having convertible com- ponents. The very general considerations adduced on pages 197-203, which are not limited in their application to gaseous bodies, suggest the hypothesis that the equations of the phases of dissipated energy of ideal gas-mixtures may apply to such gas-mixtures as have been described. It will, however, be desirable to consider the matter more in detail. J. W. (rlhhH — J^quilibriuin. of Hetero(/en€OHS Substances. 2'.\b In the first place, if we consider tlic case of a gas-mixture wliich only diifers from an ordinary ideal gas-mixture for which some of the components are equivalent in that there is perfect freedom in regard to the ti'ansformation of these components, it follows at once from the general formula of equilibrium (l) or (2) that equili- brium is only possible for such phases as we have called phases of dissipated energy, for which some of the characteristic equations have been deduced in the preceding pages. If it should be urged, that regarding a gas-mixture which has convertible components as an ideal gas-mixture of which, for some reason, only a part of the phases are actually capable of existing, we might still suppose the particular phases which alone can exist to be determined by some other principle than that of the free convertibility of the components (as if, perhaps, the case were analogous to one of constraint in mechanics), it may easily be shown that such a hypo- thesis is entirely untenable, when the quantities of the proximate components may be varied independently by suitable variations of the temperature and pressure, and of the quantities of the ultimate com- ponents, and it is admitted that 'the relations between the energy, entropy, volume, temperature, pi-essure, and the quantities of the several proximate components in the gas-mixture are the same as for an ordinary ideal gas-mixture, in which the components are not con- vertible. Let us denote the quantities of the n' proximate compo- nents of a gas-mixture A by m^, m^, etc., and the quantities of its n ultimate components by nii, nio, etc. {n denoting a number less than w'), and let us suppose that for this gas-mixture the quantities £, ?/, u, «, /J, >«j, ^2, etc. satisfy the relations characteristic of an ideal gas- mixture, while the phase of the gas-mixture is entirely determined by the values of m-i, mg, etc., with two of the quantities f, 7, w, (?,/). We may evidently imagine such an ideal gas-mixture B having n' components (not convertible), that every phase of A shall correspond yfMh one of B in the values of £, 7, v, t, p, m j , mg , etc. Now let us give to the quantities mj, mg, etc. in the gas-mixture A any fixed values, and for the body thus defined let us imagine the v-7]-e surfiice (see page 1 74) constructed ; likewise for the ideal gas-mixture B let us imagine the v-i]-£ surface constructed for every set of values of m^, m^, etc, which is consistent with the given values of m^, ixi^-, etc. i. e., for every body of which the ultimate composition would be expressed by the given values of m , , mg , etc. It follows immediately from our supposition, that every point in the v-f]-£ surface relating to A must coincide with some point of one of the v-if-e surfaces relating 236 J. W. (J^ibbs — Equilibriuiii. of Ileterof/entous Substances. to £ not only in respect to position but also in respect to its tangent plane (which represents temperature and pressure) ; therefore the «-//-£ surface relating to A must be tangent to the varioiis v-r,-e sur- faces relating to B, and therefore must be ai] envelop of these sur- faces. P'roni this it follows that the points which represent phases common to both gas-mixtures must represent the phases of dissipated energy of the gas-mixture B. The properties of an ideal gas-mixture which are assumed in regard to the gas-mixture of converti1)le components in the above demonstration are expressed by equations (277) and (278) with the equation e:=:£A'\>",f-^"'iE,). (824) It is usual to assume in regard to gas-mixtures liaving convertible components that the convertibility of the components does not affect the relations (277) and (324). The same cannot be said of the equa- tion (278). But in a very important class of cases it will be sufficient if the applicability of (277) and (324) is admitted. The cases referred to are those in which in certain phases of a gas-mixture the compo- nents are convertible, and in other phases of the same proximate composition the components are not convertible, and the equations of an ideal gas-mixture hold true. If there is only a single degree of convertibility between the com- ponents, (i. e., if only a single kind of conversion, with its reverse, can take place among the components,) it Avill be sufficient to assume, in regard to the phases in which conversion takes place, the validity of equation (277) and of the following, which can be derived from (324) by differentiation, and comparison with equation (11), which expresses a necessary relation, \t d )} —p dv - 2j (c^m^) dt] „. = 0.* (325) We shall confine our demonstration to this case. It will be observed that the physical signification of (325) is that if the gas-mixture is subjected to such changes of volume and temperature as do not alter its proximate composition, the heat absorbed or yielded may be cal- culated by the same formula as if the components were not conver- tible. Let us suppose the thermodynamic state of a gaseous mass J/, of such a kind as has just been described, to be varied while within the limits within which the components are not convertible. (The quan- tities of the proximate components, therefore, as well as of the ulti- * This notation is intended to indicate that ?;i|, m.^, etc. are regarded as constant J. W' . Glbbs — Kqa 11 lb I'll (III of Jleteroi/aneoiis Sabst<incef<. 2;{7 mate, are supposed constant). If we vise the same metliod of geome- trical representation as before, the point representing- the vohime, entropy, and energy of the mass will describe a line in the n-ij-f: sur- face of an ideal gas-mixture of inconvertible components, the form and position of this surface being determined by the proximate comi)0- sition of 31. Let us now suppose the same mass to be carried beyond the limit of inconvertibility, the variations of state after passino- the limit being such as not to alter its proxinuxte composition. It is evident that this will in general be possible. Exceptions can only occur when the limit is formed liy phases in which the proximate composition is uniform. The line traced in the region of convertibility must belong to the same »-?/-£ surface of an ideal gas-mixture of in- convertible components as before, continued beyond the limit of inconvertibility for the components of 31, since the variations of volume, entropy and energy are the same as would be possible if the components were not convertible. But it must also belong to the v-7]-8 surface of the body J/, which is here a gas-mixture of conver- tible components. Moreover, as the inclination of each of these surfaces must indicate the temperature and pressure of the phases through which the body passes, these two surfaces must be tangent to each other along the line which has been traced. As the y-;/-£ surface of the body 31 in the region of convertibility must thus be tangent to all the surfaces representing ideal gas-mixtiires of every possible proximate composition consistent with the ultimate composi- tion of 31, continued beyond the region of inconvertibility, in which alone their form and position may be capable of experimental demon- stration, the former surface must be an envelop of the latter sui-faces, and therefore a continuation of the surface of the phases of dissipated energy in the region of inconvertibility. The foregoing considerations may give a measure of a priori prob- ability to the results which are obtained by applying the ordinary laws of ideal gas-mixtures to cases in which the components are con- vertible. It is only by experiments upon gases in phases in which their components are convertible that the validity of any of these results can be established. The very accurate determinations of density which have been made for the peroxide of nitrogen enable us to subject some of our equa- tions to a very critical test. That this substance in the gaseous state is properly regarded as a mixture of different gases can hardly be doubted, as the proportion of the components derived from its density on the supposition that one component has the molecular formula 238 tf. W. Gibbs — Equilihrintn of Heterogeneous Substances. NOg and the other the formula N^O^ is the same as that derived from the depth of the color on the supposition that the absorption of light is due to one of the components alone, and is proportioned to the separate density of that component.* MM. Sainte-Claire Deville and Troostf have given a series of deter- minations of what we shall call the relative densities of peroxide of nitrogen at various temperatures under atmospheric pressure. We use the terra relative density to denote Avhat it is usual in treatises on chemistry to denote by the term density, viz., the actual density of a gas divided by the density of a standard perfect gas at the same pressure and temperature, the standard gas being air, or more strictly, an ideal gas which has the same density as air at the zero of the centigrade scale and the pressure of one atmos])here. In order to test our equations by these determinations, it will be convenient to transform equation (320), so as to give directly the relation between the relative density, the pressure, and the temperature. As the density of the standard gas at any given temperature and P pressure may by (263) be expressed by the formula -^—, the relative density of a binary gas-mixture may be expressed by Now by (263) a t 7>= (m, +^2)-^. (326) ^ ^ pv a^ 7n^ -\- ao ni^ =: — . (327) By giving to ^3 and m^ successively the value zero in these equa- tions, we obtain O I ^2 where D-^ and Z>2 denote the values of D when the gas consists wholly of one or of the other component. If we assume that JJ,=2IJ„ (329) we shall have From (326) we have «i = 2a2. (330) m , -{- n/^ :=. JJ , *Salet, "Sur la coloration du peroxyde d'azote," Comptes Eendiis, vol. Ixvii, p. 488. f Comptes Rendus, vol. Ixiv, p. 237. J. TTT a ihhs— Equilibrium of Heterogeneous Substances. 239 and from (327), by (828) unci (;i80), 2 m, -{-»i.,z= I>J~-=i2J)- , whence m, = (D,~lJ)i^-^, (331) m, = 2(7>-y>,)f3- (•'^^2) By (327), (331), and (332) we obtain from (320) los^~ r.\ = log^ . (333) ^2 (Z> - i>i) a, a^ a„ ° « ^ ' This formula will be more convenient for purposes of calculation if we introduce common logarithms (denoted by log,g) instead of hyperbolic, the temperature of the ordinary centigrade scale t, instead of the absolute temperature t, and the pressure in atmospheres p„t instead of p the pressure in a rational system of units. If we also add the logarithm of a, to both sides of the equation, we obtain •°g.» -^(i-S-x = ^ + 1 '"=■» <' +^"^" - ^3- (^•^*) where A and C denote constants, the values of which are closely con- nected with those of A and 0. From the molecular formula? of peroxide of nitrogen NO^ and NgO^, we may calculate the relative densities 14 + 32^^^^^ _ ^ .^ ^^^^^ j^ — !^jhlf ,0691 r= 3.178. (335) 1 2 ' - 2 ^ ' The determinations of MM. Deville and Troost are satisfactorily represented by the equation iogio 2 (i> - 1.589) ^,+ 273' ^ ^ which o'ives i)= 3.178+ (y - VW(3.178H-0) 3118.6 , where log ^o^J= 9.47056 - f_^^ - logi oP<u- In the first part of the following table are given in successive col- umns the temperature and pressure of the gas in the several experi- ments of MM. Deville and Troost, the relative densities calculated from these numbers by equation (336), the relative densities as observed and the difference of the observed and calculated relative 240 ./ W. Gihbs — Equilihrimn of Heterogeneous S^ihstanees. densities. It will be observed that these differences are quite small, in no case reaching .03, and on the average scarcely exceeding .01. The significance of such correspondence in favor of the hypothesis by means of which equation (336) has been established is of course diminished by the fact that two constants in the equation have been determined from these experiments. If the same equation can be shown to give correctly the relative densities at other pressures than that for Avhich the constants have been determined, such correspon- dence will be much more decisive. t. Pat D calculated by eq. (336). D observed. diff. Observers. 26.7 2.676 2.65 -.026 D. & T. 35.4 2.524 2,53 + .006 D. & T. 39.8 2.443 2.46 + .017 D. & T. 49.6 2.256 2.27 + .014 D. & T. 60.2 2.067 2.08 + .013 D. & T. 70.0 1.920 1.92 .000 D. & T. 80.6 1.801 1.80 -.001 D. & T. 90.0 1.728 1.72 -.008 D. & T. 100.1 1.676 1.68 + .004 D. & T. 111.3 1.641 1.65 + .009 D. & T. 121.5 1.622 1.62 -.002 D. & T. 135.0 1.607 1.60 -.007 D. & T. 154.0 1.597 1.58 -.017 D. & T. 183.2 1.592 1.57 -.022 D. & T. 97.5 1.687 97.5 iHf? 1.631 1.783 + .152 P. & W. 24.5 2.711 24.5 \nn 2.524 2.52 -.004 P. & W. 11.3 2.891 11.3 '442 6 5 2.620 2.645 + .025 P. & w. 4.2 2.964 4.2 3 S 4 3 8 2.708 2.588 -.120 P. & w. Messrs. Playfair and Wanklyn have published* four determinations of the relative density of peroxide of nitrogen at various temperatures when"diluted with nitrogen. Since the relations expressed by equa- tions (319) and (320) are not affected by the presence of a third gas which is different from the gases O^ and G2 (to which m.^ and ui^ relate) and neutral to them, (see the remark at the foot of page 233), — provided that we take^j) to denote the pressure which we attribute to the gases (x, and (t2i^- ^-i the total pressure diminished by the pressui-e which the third gas would exert if occupying alone the same space at the same temperature, — it follows that the relations expressed for * Transactions of the Royal Society of Edinburg, vol. xxii, p. 441. ./ W. Gihhs — EqxIJlhrhoii. of Heterogeneous Suhstancex. 241 peroxide of nitrogen by (333), (334), and (336) will not be aliected by the presence of free nitrogen, if the pressure expressed bv ^> or jt),„ and contained implicitly in the symbol IJ (see equation (320) l)y which D is defined) is understood to denote the total pressure dimin- ished by the pressure due to the free nitrogen. The determinations of Playfiiir and Wanklyn are given in the latter part of the above table. The pressures given are those obtained by subtracting the pressure due to the free nitrogen from the total pi-essure. We may suppose such reduced pressures to have been used in the reduction of the observations by which the numbers in the column of observed relative densities were obtained. Besides the relative densities calculated by equation (336) for the temperatures and (reduced) pressures of the observations, the table contains the relative densities calculated for the same tem])eratures and the pressure of one atmos- phere. The reader will observe that in the second and third experiments of Playfair and Wanklyn there is a very close accordance between the calculated and observed values of D, while in the second and fourth experiments there is a considerable diiference. Now the weight to be attributed to the several determinations is very diifer- ent. The quantities of peroxide of nitrogen which were used in the several experiments were respectively .2410, .5893, .3166, and .2016 grammes. For a rough approximation, Ave may assume that the probable errors of the relative densities are inversely proportional to these numbers. This would make the probable error of the first and fourth observations two or three times as great as that of the second and considerably greater than that of the third. We must also observe that in the first of these experiments, the observed relative density 1.783 is greater than 1.687, the relative density calculated by equation (336) for the temperature of the experiment and the pres- sure of one atmosphere. Now the number 1.687 we may regard as established directly by the experiments of Deville and Troost. For in seven successive experiments in this part of the series the calculated relative densities difter from the observed by less than .01. If then we accept the numbers given by experiment, the efiect of diluting the gas with nitrogen is to increase its relative density. As this result is entirely at variance with the facts observed in the case of other gases, and in the case of this gas at lower temperatures, as appears from the three other determinations of Playfair and Wanklyn, it cannot possibly be admitted on the strength of a single Trans. Conn. Acad., Vol. III. 31 Mat, 187r,. 242 J. W. Gibbs — Equilibrium of Heterogeneous Substances. observation. The first experiment of this series cannot therefore properly be used as a test of our equations. Similar considerations apply with somewhat less force to the last experiment. By compar- ing the temperatures and pressures of the three last experiments with the observed relative densities, the reader may easily convince himself that if we admit the substantial accuracy of the determina- tions in the two first of these experiments (the second and third of the series, which have the greatest weight), the last determination of relative density 2.588 must be too small. In fact, it should evidently be greater than the number in the preceditig experiment 2.645. If we confine our attention to the second and third expei'iments of the series, the agreement is as good as could be desired. Nor will the admission of errors of .152 and .120 (certainly not large in deter- minations of this kind) in the first and fourth experiments involve any serious doubt of the substantial accuracy of the second and third, when the difference of weight of the determinations is considered. Yet it is much to be desired that the relation expressed by (336), or with more generality by (334), should be tested by more numerous experiments. It should be stated that the numbers in the column of pressures are not quite accurate. In the experiments of Deville and Troost the gas was subject to the actual atmospheric pressure at the time of the experiment. This A^aried from 747 to 764 millimeters of mercury. The precise pressure for each experiment is not given. In the ex- periments of Playfair and Wanklyn the mixture of nitrogen and peroxide of nitrogen was subject to the actual atmospheric pressure at the time of the experiment. The numbers in the column of pres- sures express the fraction of the whole pressure wliich remains after substracting the part due to the free nitrogen. But no indication is given in the published account of the experiinents in regard to the height of the barometer. Now it may easily be shown that a varia- tion of j^^xs ill t.he value of p can in no case cause a variation of more than .005 in the value of D as calculated by equation (336). In any of the experiments of Playfair and Wanklyn a variation of more than 30""" in the height of the barometer would be necessary to produce a variation of .01 in the value of D. The errors due to this source cannot therefore be very serious. They might have been avoided altogether in the discussion of the experiments of Deville and Troost by using instead of (336) a formula expressing the relation between the relative density, the temperature, and the actual density, as the reciprocal of the latter quantity is given for each ex- J. W. (rlhbs — I'JquUlbrluiii of Heterogeneous Substances. 24.3 perimeiit of this series. It seemed best, however, to make a triHiiio; sacrifice of accxiracy for the sake of simplicity. It might be thought that the experiments under discussion would be better represented by a formula in which the term containing log t (see equation (333)) was retained. But an examination of the figures in the table will show that nothing important can be gained in this respect, and there is hardly sufticient motive for adding another term to the formula of calculation. Any attempt to determine the real values of A, B', and C in equation (333), (assuming the absolute validity of such an equation for peroxide of nitrogen,) from the ex- periments under discussion would be entirely misleading, as the reader may easily convince himself. From equation (336), however, the following conclusions may deduced. By comparison with (334) we obtain . , ^' C ^ 311S.6 A + — log.o « - 7 = 9-47056 ^, which must hold true approximately between the temperatures 11*^ and 90'\ (At higher temperatures the relative densities vary too slowly Avith the temperatures to afibrd a critical test of the accuracy of this relation.) By diiFerentiation we obtain Jlf_S'C_ 3118.6 a^t *" ¥~ W~' where 31 denotes the modulus of the common system of logarithms. Now by comparing equations (333) and (334) we see that MC C C = = .43429 — . Hence B'i-^ C= 7181 «2 = 3590 «j, which may be regarded as a close approximation at 40'-' or 50^', and a tolerable approximation between the limits of temperature above mentioned. Now B' t + C represents the heat evolved by the con- version of a unit of NOg into NgO^ under constant pressure. Such conversion cannot take place at constant pressure without change of temperature, which renders the experimental verification of the last equation less simple. But since by equations (322) B' = B+a^ - rt2 = ^ + i«i, we shall have for the temperature of 40*^' Bt-\- C;=3434a,. Now B t -\- (J reiH-esents the decrease of energy when a unit of NOg is transformed into NgO^ without change of temperature. It therefore 244 J. W. Gihbs — EquUihrium of Heterogeneous Substances. represents the excess of the heat evolved over the work done by external forces when a mass of the gas is compressed at constant temperatnre until a unit of NO 3 has been converted into NgO^. This quantity will be constant if J3 =zO, i. e., if the specific heats at constant volume of NO^ and N2O4 ^i"e the same. This assumption would be more simple from a theoretical stand-jjoint and perhaps safer than the assumption that B' = 0. li B =1 0, B' = a^. If we wish to embody this assumption in the equation between Z>, p^ and t, we may substitute 2977 4 6.5228 + log, „ {t, + 273) ^ j^i^^ for the second member of equation (336). The relative densities calculated by the equation thus modified from the temperatures and pressures of the experiments under discussion will not diflTer from those calculated from the unmodified equation by more than .002 in any case, or by more than ,001 in the first series of experiments. It is to be noticed that if we admit the validity of the volumetrical relation expressed by equation (333), which is evidently equivalent to an equation between p, t, y, and ni (this letter denoting the quan- tity of the gas without reference to its molecular condition), or if we admit the validity of the equation only between certain limits of temperature and for densities less than a certain limit of density, and also admit that between the given limits of temperature the specific heat of the gas at constant volume may be regarded as a constant quantity when the gas is sufficiently rarefied to be regarded as con- sisting wholly of NO2, — or, to speak without reference to the m.olecu- lar state of the gas, when it is rarefied until its relative density D approximates to its limiting value Z>,, — we must also admit the validity (within the same limits of temperature and density) of all the calorimetrical relations which belong to ideal gas-mixtures with convertible components. The premises are evidently equivalent to this, — that we may imagine an ideal gas with convertible components such that between certain limits of temperature and above a certain limit of density the relation between p, t, and v shall be the same for a unit of this ideal gas as for a unit of peroxide of nitrogen, and for a very ^reat value of 0 (witliin the given limits of temperature) the thermal capacity at constant volume of the ideal and actual gases shall be the same. Let us regard t and v as independent variables ; we may let these letters and p refer alike to the ideal and real gases, but we must distinguish the entropy ?/' of the ideal gas from the entropy tf of the real gas. Now by (88) J. W. Gibbs — Equilibrimn of Heterogeneous Substances. 245 dn dp therefore d dt] d dfj d dp d~p dv dt ~ dt dv ~ dt dt ~ dt^' (338) Since a similar relation will hold true for //, we obtain d dy] d dtf d^ ~dt~d^ ~dV (^'^^^ which must hold true within the given limits of temperature and density. Now it is granted that dt - dt ^^^^^ for very great values of o at any temperature Avithin the given limits, (for the two members of the equation represent the thermal capacities at constant volume of the real and ideal gases divided by ^,) hence, in virtue of (339), this equation must hold true in general within the given limits of temperature and density. Again, as an equation like (337) will hold true of ?/', we shall have drj dii dv dv' ^ ' From the two last equations it is evident that in all calorimetrical relations the ideal and real gases are identical. Moreover the energy and entropy of the ideal gas are evidently so far arbitrary that we may suppose them to have the same values as in the real gas for any given values of t and v. Hence the entropies of the two gases are the same within the given limits ; and on account of the necessary relation ds-=. t dt] — p dv, the energies of the two gases are in like manner identical. Hence the fundamental equation between the energy, entropy, volume, and quantity of matter must be the same for the ideal gas as for the actual. We may easily form a fundamental equation for an ideal gas-mix- ture with convertible components, which shall relate only to the phases of equilibrium. For this purpose, we may use the e(piations of the form (312) to eliminate from the equation of the form (273), which expresses the relation ' between the pressure, the temperature, and the potentials for the proximate components, as many of the potentials as there are equations of the former kind, leaving the 246 J. W. Gibbs — Equilibrium of Heterogeneous Substances. potentials for those components which it is convenient to regard as the ultimate components of the gas-mixture. In the case of a binary gas-mixture with convertible components, the components will have the same potential, which may be denoted by //, and the fundamental equation will be Ci+ a, // — -£/, c.2-i-a.2 /U — E2 p = a^L^t e -\-a.^L^t e ^ , (342) where (343) From this equation, by differentiation and comparison with (98), we obtain c, ii-E, -=^c, + a,-~--jt e + -^2 ( <'2 + «2 -^ ~t U'e , (244) c^ jLi — Ef c-i fi — E2 - = L^t e -{- L^t e (345) From the 'general equation (93) with the preceding eqiiations the following is easily obtained, — c^ [i—E^ Cj^ li — Ej --L^{c^t^E^)t"" e "'^ ■^L^{c^t^E.^)f' e "-'^ . (346) We may obtain the relation between jo, t, v, and ni by eliminating pi from (342) and (345). For this purpose we may proceed as follows- From (342) and (345) we obtain Cj -t-a, fJ-—Ei Cj + Ct2 /^ — -5*2 a^t p — (a^ - Uo) E^t ' e ' ; (348) and from these equations we obtain a, log Ip-a^t^-^j-a^log ia,t"^-p] = {a^ - "3) log (a,— aj E — 7^ + a,logii-«2log^-'3+(^'2-''2+^«i-«2)log« --r-^' (349) J. IK Gihha — Equilibrium of TTelerogeneous Substances. 247 (In the particular case when a, = 2 «„ tliis equation will he e(iuiva- lent to (333)). By (347) and (348) we may easily eliminate i^i from (346). The reader will observe that the relations thus deduced from the fundamental equation (342) without any reference to the different components of the gaseous mass are equivalent to those which relate to the phases of dissipated energy of a binary gas-mixture with com- ponents which are equivalent in substance but not convertible, except that the equations derived from (342) do not give the quantities of the proximate components, but relate solely to those properties which are capable of direct experimental verification without the aid of any theory of the constitution of the gaseous mass. The practical apj^lication of these equations is rendered more simple by the fact that the ratio a^'.a^ will always bear a simple relation to unity. When a, and «2 are equal, if we write a for their common value, we shall have by (342) and (345) 2? V =1 a m t, (3.50) and by (345) and (346) m C2~C| E. —E2 at (.351) ij -j- 2^2^ ^ By this equation we may calculate directly the amount of heat required to raise a given quantity of the gas from one given tempera- ture to another at constant volume. The equation shows that the amount of heat will be independent of the volume of the gas. The heat necessary to produce a given change of temperature in the gas at constant pressure, may be found by taking the difference of the values of J, as defined by equation (89), for the initial and final states of the gas. From (89), (350), and (351) we obtain Cj—Cj, El— Est Z _ Z^{cit-\-at+B^) + L2{c2t + at+E^)t "" e "^ ^ .g^^) L^+L„t e By differentiation of the two last equations we may obtain directly the specific heats of the gas at constant volume and at constant pres- sure. The fundamental equation of an ideal ternary gas-mixture with a single relation of convertibility between its components is 248 J. JV. Gibbs — Equilibrium of Heterogeneous Substances. Hi—Ci — a^ c, +a| n^—E^ a, , a, ttjt p = a^ e t e a., ^ O'i o,>t 4- r/g e ' t e + «3 e t ^ e , (353) where Aj and X.^ have the same meaning as on page 230. {I'o he continued.) ERRATA. Page 167, formula (168), for m, read fi^. " formula (169), for to,, . . . m„_j read/z,, . . . //«— i- Page 239. formula (333), for— read _iL. t ttat VI. The Hydroids op the Pacific Coast of the United States, SOUTH OF Vancouver Island. With a Report upon those IN THE Museum of Yale College. Bv S. F. Clari Read Jan. 19. 18 T 6. The Museum is indebted for its collection of Californian Hydroids chiefly to Pro!'. D. C. Eaton, who has presented during the last two or three years, a large number of specimens, that were received by him with dried algtt? from that coast. They were collected and sent to him by Dr. C. W. Anderson, Santa Cruz, Cal. ; Dr. L. N. Dimmick, Santa Barbara, Cal. ; Mi-s. Ell wood Cooper, Santa Barbara, Cal. ; and ^Nliss jNIitchell of Vancouver Island. All the specimens received from these sources were collected in tide-pools along the shore or attached to algae, washed in froui deeper water. A few alcoholic specimens have also been received from San Diego, Cal., collected on the piles of the Avharves and along the shore, by Dr. E. Palmer, and a fine specimen of Plumularia setacea was dredged in six to eight fathoms, oil' San Diego, by Mr. Henry Hemphill. Some of the species, including most of the Sertnlarid(e, do not seem to be injured by being dried, but others, as the Campannlaridce, ai'e usually rendered useless for description. The specimens of the two species of < ■ampanularia described below are unusually well preserved, both hydrothecae and gonothecae being in good condition. There has been very little published on the hydroids of the western coast of North America, up to tlie present time. In 1857 Dr. Trask* described and figured nine new species of Zoophytes from the Bay of San Francisco and adjacent localities. Five of these are Bryozoa ; the remaining four represent three genera of the family Sertularidie, as follows: Sertularia anguind, S./ureata, Sertu- lareUa turgida and HydvaUmania Franciscana, all of which, so far as I am aware, are peculiar to that coast ; unless indeed the last named species prove to be identical with H. falcata of Europe, * Proceedings of the California Academy of Natural Sciences, vol. i, March, 1357. Dr. J. B. Trask. Trans. Conn. Acad., Vol. III. 1 June, 1876. 250 S. M Clarh — Hydroids of the Pacific Coast. Africa and New England. In 1860 Andrew Murray* described and figured five species from the Californian coast, of which three are new, and the other two, Sertidaria labrata and Plumalaria gracilis^ are synonymous with two of Trask's species, viz : Sertularia anguina and Plumularia Franciscana. Mr. Alexander Agassizf in 1865 described seven species and recorded seven others from the Bay of San Francisco ; and he had three of the same from the Gulf of Georgia, W. T. Five species were also mentioned by him from the North Pacific. Two of these five northern species, Bougainvillia Mertensii Ag. and Cotidina Greenei A. Ag., are also found at San Francisco. The latter species having also been collected at Santa Barbara, Cal., has the wide j-ange of nearly three thousand miles upon our western coast. Professor Allman mentions having found sixteen species in a collection from the Californian coast, siil)mitted to him for examination ; two of them, Lafoea. dumosa and Sertularia pumila, are common on the European and New England coasts, and the former species is also recorded from South Africa. The collection in the Museum of Yale College contains twelve species and one variety. Of these four are new ; nine are recorded only from the Pacific coast of North America, as yet; and three, Halecium. tenellum^ ISertularia argentea and Phmiidaria setacea, are also common on the European shores; the first two of these have also been found on the New England coast, from Maine to Long Island Sound. The most common form on the Californian coast is the showy Aglaophenia struthionides, which is apparently as abundant there as Sertularia argentea and S. pumila are upon our eastern shores, for it forms the bulk of every package sent to lis from the western coast. The folloAving table gives a list of all the Hydroids known on the western coast of the United States, from Vancouver's Island to San Diego, with the range of tlie different species and the names of some of the collectors. List of Hydroida known to occur between San Diego and Vancouver Island. Coryne rosaria A. Ag. Bay of San Francisco, Cal. (A. Agassiz). Tubularia elegans Clark. San Diego, Cal. (Dr. 15. Palmer). Thamnocnidia tubiilaroides A. Ag. Bay of San Francisco, Cal. (A. Agassiz). * Tlie Annals and Magazine of Natural History, Series 3, No. XXVIII, April, 1860. Descriptions of new species of Hydroids from the Californian Coast. By Andrew Murray. \ Illustrated Catalogue of the Museum of Comparative Zoology. No. II. North American Acalephre. By Alexander Agassiz. 1805.^ ^S". F. Clark — Ilydroids of the Pacific Coast. 251 Par3'plia microcephala A. Ag. Bimeria gracilis Clark. Bougainvillia Mertensii Agassiz. Eudendrium, sp. Campauularia everta Clark. Campanularia fusiformis Clark. Campanularia cyliudrica Clark. Laomedea rigida A. Ag. Laomedea Pacifica A. Ag. Lafoea dumosa Sars. Halecium tenellum Hincks. Sertularia angiiina Trask. Sertularia anguina, var robusta Clark. Sertularia argeutea E. and S. Sertularia pumila Linn. Sertularia Greenei Murray. Bay of Sau Francisco, Cal. (A. Agassiz). Sau Diego, Cal. (E. Palmer). Bay of San Francisco, Cal. (A. Agassiz). Santa Cruz, Cal. (C. W. Anderson). San Diego, Cal. (H. Hemphill), to Vancouver Island (J. M. Dawson). Vancouver Island (J. M. Dawson). Santa Cruz, Cal. (C. W. Anderson). Bay of San Francisco, Cal. (A. Agassiz). Gulf of Georgia (A. Ag.) to Bay of San Francisco (A. Agassiz). San Diego, Cal. (Dr. E. Palmer). Santa Cruz, Cal. (C. "W. Anderson), to Van- couver Island (J. M. Dawson). San Diego, Cal. (H. Hemphill), to Vancouver Island (J. M. Dawson). Santa Barbara, Cal. (Mrs. EUwood Cooper). Sertularia furcata Trask. Sertularia corniculaia Murray. Sertularella turgida Clark (Trask). Plumularia setacea Lamarck. Santa Barbara, Cal. (Mrs. P]llwood Cooper), to Vancouver Island (J. M. Dawson). San Diego (Dr. E. Palmer), to Bay of San Francisco (J. B. Trask). Bay of San Francisco (A. Murray). San Diego, Cal., to Vancouver Island (J. M. Dawson). San Diego, Cal. (Dr. E. Palmer), to Van- couver Island (J. M. Dawson). Aglaophenia struthionides Clark (Murray). San Diego, Cal. (D. C. Cleveland), to Van- couver Island (Miss Mitchell). This list of twenty-four species is very small compared with that of the eastern coast, from Maine to New York, the fauna of the latter region containing five times as many species as that of the former, notwith- standing that the i-egion included on the western coast is over thirteen hundred miles in length, while that of the New England coast is only about eight hundred. It should be borne in mind however that most of the collecting on the Pacific coast has been done along the shore, the dredge having been little used, and there is little doubt that when the fauna has been more thoroughly investigated the number of Hydroids may be at least doubled. Such a variety as exists on the New England coast can hardly be expected from our Pacific shores south of Vancouver Island, for the waters there do not afford the same diversity in temperature. 252 S. F. Clark — Ilydroids of the Pacific Coast. Bimeria (?) gracilis, sp. nov. riate XXXVIII, figure 3. Stems clustered, rooted by a creeping stolon, erect, simple, delicate, not divided by distinct joints, thickly branched ; branches suberect, the larger ones reaching to the end of the stem and resembling the main stalk, the smaller ones bear but one or two hydranths and are also unjointed; perisarc extending over the hydranths and partially covering the tentacles, annulated at the base of each branch and branchlet. Sporosacs developed from the hydrophyton, a single one at the base of each hydranth-bearing branchlet, oval or ovate, sup- ported by a short peduncle consisting of one or two annulations. Hydranths large, tapering uniformly from the distal end to the base, provided with about ten or twelve tentacles and with a large, rounded or slightly conical proboscis. Height of best specimen, 55"'"'. Collected on the jtiles of wharves at San Diego, Cal., by Dr. E. Palmer, 18'75. Our specimens were not in a good condition when they arrived, having been crowded in a tin can with many other things, which pressed them all out of shape, and the quantity of alcohol not being sufficient to preserve so much animal matter, the hydi-oids suffered considerably; the hydranths and sporosacs especially were in a very worn and mutilated state. It is not easy to determine just how far the perisarc extends upon the hydranth, but it certainly covers tlie body of the latter, and it must, I think, be developed over a portion of the tentacles, for after soaking them in a dilute solution of caustic potash for forty-eight hours the tentacles still retained their normal position, nor did they show any decrease in size. The potash seemed to act very slowly, for after being in the warm solution forty-eight hours the hydranths were not entirely dissolved out. The fact of the tentacles being unaftected would seem to indicate that they are entirelv protected by chitin, but tentacles so protected would be of little or no use to the animal, and I think it more j)rol)able that the distal portions are free and may be contracted into the basal covering. It is impossible to determine from our specimens how the tentacles are held, whethei- in a single erect verticil as in Garveia or with each alternate tentacle depressed, as in Bimeria vestita of Wright. With such imi)erfect data I feel some doubt about placing this species in the genus Bimeria, and only do so provisionally. >S'. F. Clark — Ilydroids of the Pacific Coast. 253 Tubularia elegans, sp. nov. Plate XXXVIII, figure 2. Stems clustered, rooted by a creeping stolon, erect, unbranched, more or less annulated at intervals toward the base. Hydranths large, with about thirty tentacles in the proximal set and twenty to twenty-four in the distal. Gonophores borne in clusters just inside the proximal tentacles, twelve to twenty in a cluster, each of the larger ones crowned with four conical tubercles. Height of tine.st specimen, 75""". Collected on the piles of the wharf at San Diego, by Dr. E. Palmer 1875. Intermingled with it and often attached to it were numerous shoots of Blmeria. Many of the young had attached themselves to the parent stalk, giving at first sight the appearance of branching stems. , The specimens from which this species is described were crowded in the same can with the Bimeria described above, and are in the same dilapidated condition. There is a Tubularian, Tha/mnocnidla tuhular- oides, from the Bay of San Francisco, described by A. Agassiz (Cat. of N. A. Acalephje, p. 196), which he says "is readily distinguished from its eastern congeners by the stoutness of the stem and large size of the head." The description is a very meagi-e one, but from these two characters I conclude that it must be distinct from T. elegans, for the latter spetaes has neither a stouter stem nor larger head than Tham- nocnidia spectahilis of the New England coast. Eudendrium, sp. Plate XXXVIII, figure 1. We have also received from the California coast the perisarc or chitinous portion of what I take to be a species of Eudendrium. Stems stout, erect, dark horn color, strongly annulated throughout, rather sparingly branched ; branches sub-erect, springing from all sides of the stem and much divided. Hydranths borne at the ex- tremity of the short ramuli. The entire perisarc is strongly ringed, giving it a close resemblance to the trachete of an insect. Height of largest specimen, 80"'"'" Santa Cruz, Bay of Monterey, Cal., — Dr. C. W. Anderson. Campanularia everta, sp. nov. Plate XXXIX, figure 4. Stems rather stout, arising at intervals from the creeping stolon, with two annulations at the base of the hydrothecte, the lower one 254 S. F. Clark— Hydr Olds of the Pacific Coast. smaller than the upper; the remainder of the stem has a wavy outline or is slightly annulated, Hydrothec^ broadly campanulate, not deep, tapering more or less gradually from the distal end to the base, the rim strongly everted and bearing about fifteen rather shallow teeth. Gonothecje, large, turgid, nearly cylindrical, tapering a little at the base, borne on short, stout peduncles and with the aperture terminal, small and cylindrical. Found creeping on an Alga from San Diego, Cab, — H. Hemphill. This is a very pretty form and may readily be distinguished by the broad hydrothecae with their strongly everted, toothed rims. The peculiar shape of the gonothecte is also very characteristic. Campanularia cylindrica, sp. nov. Plate XXXIX, figures 1-r'. Stems are simple, unbranched pedicels, of very variable length, more or less annulated over the entire length and with a single well- marked ring at the base of the hydrothecje, rooted by a creeping, twisted stolon. Hydrothecte campanulate, nearly cylindrical, taper- ing but very slightly toward the base, varying greatly in depth, rim armed with about fifteen very shallow, sharply pointed teeth. The gonothecse also show considerable variation in size, there being occa- sionally one or two which are at least twice the size of the ordinary form; they are subfusiform, tapeiing sliglitly more toward the proxi- mal than the distal end, supported on short pedicels with one or two annulations. Campanularia fusiformis, sp. nov. Plate XXXIX, figures 2-2-. Hydrocaulus simple, creeping, bearing the pedicels at irregular intervals; ])edicels of variable length, usually two or three times the length of the hydrothec;^, never more than six times their length, with a more or less w^avy outline. Hydrotheca? small, deeply campanulate, tapering at the base, rim ornamented witli about twelve stout, shallow, acute teeth, a single distinct annulation at the base. GonotheciB small, fusiform, constricted at both ends, sessile, aperture small, terminal. Vancouver Island, — J, M. Daw^son. Found growing on Sertularia angidna var. rohusta. This species is closely allied to C. cylindrica of the Californian coast from which it may be distinguished by the size of the hydro- S. F. Clark — Sydroids of the Pacific Coast. 255 thecffi and by their shajjc, not being rounded at the base ; by the form of tlie gonotliectp, wliicli are sessile and liave a circular terminal aperture. Found creeping on the old stems of a MtdendriuniWke form, taken at Santa Cruz, Cal., Bay of Monterey, by Dr. C. W. Anderson. The variation in the lengtli of the stems is very great ; sometimes they are aboiit equal to the length of the hydrothecjje, and again they will be five or six times that length. The stolon is quite uniformly twisted and is at least twice the diameter of the stems. Halecium tenellum. Hincks. Halecium tenellum Hincks, Annuls and Map;, of Nat. Hist., 3, VIII. 252, pi. VI. Plate XXXIX, figure 5. Some very good specimens of this delicate species have been received from San Diego. There were no gonothecfe but the hydro- some is so exactly similar to our New England specimens and to the figure and description of Hincks that I do not hesitate to call it the same. A glance at our figure will show how exactly it corresponds. Found parasitic on a species of Bimeria, collected on the piles of wharves, San Diego, Cal., — Dr. E. Palmer, 18Y5. Sertularia anguina Trask. Sertularia anguina Trask, Proc. Cal. Acad. Nat. Sci., 112, Plate V, fig. 1. March 30, 1857. Sertularia Inhrata Murray, Ann. and Mag. for April, 1860, 250, Plate XI, fig. 2. Plate XL, figures 1, P, 2. Stems clustered, simple, erect, straight from the proximal end to the first branch, above the first branch flexuous, becoming more and more so toward the distal end, sparingly branched, divided by trans- verse joints into short internodes, those below the first branch bearing a single pair of nearly opposite hydrothecre, while those above the first branch have three hydrotheese and give origin to a single branch ; branches regularly alternate, ascending, slightly curved, mostly short, a few have a much larger growth and exactly imitate the main stems in every particular; color corneous. Hydrotheca3 nearly opposite, somewhat flask-shaped or tapering evenly to the distal end with- out any constriction or flask-shaped neck ; aperture usually entire, slightly oblique, facing toward the stem, or with the outer margin much more produced than the inner and in some cases showing a dis- 256 S. P. Clark — Itydroids of the Pacific Coast tinctly sinuous outline. Gonothecse unknown. Height of largest specimen, 75"'"'. Plentiful on a large species of Mytilus from Mon- terey, Tomales Pt., Punta Reyes, and on old shells, Bay of San Fran- cisco (Dr. Trask) ; Santa Cruz, — Dr. Anderson; Vancouver Island, — J. M, Dawson. Our specimens of S. anguina agree so closely with Hinck's descrip- tion and figures of S. filicula that I cannot separate the trophosomes, but the gonosomes being unknown, I prefer to let the species remain distinct rather than to unite them on such incomplete data. Murray also noticed the similarity to *S'. filicula although he only possessed " a minute portion without vesicles." Dr. Trask says of his specimens, " Their affinity is witli that of 8. fallax of Johnston more nearly than with any other species with which I am acquainted." He could not have known (^f ^i. filicula at that time or he would at once have noticed the much closer resem- blance to that form. Pie also says that his specimens have four hydrothecie between each pair of branchlets, while ours have but three: one pair opposite each othei- and one odd cell in the axil of the branch. His description and figure agree so well, however, in evei'y other respect that I am inclined to regard this as an error of observa- tion. This description has been made from specimens which were dried before they were sent to iis and have since been soaked out in warm water. The perisarc being very stout and durable I do not think that it can have changed to any great extent. Sertularia anguina, variety robusta Clark. Plate XL, figures 3, 4, 5. The variety robusta differs from the ordinary form in having a stouter stem, larger hydrotheca% longer pinnjie, and in being in every way a much larger and stouter form. The mode of growth, the branching, the shape and arrangement of the hydrothecae are the same as in the normal form. Gonothecae borne on the pinnae, more or less fusiform, arising from just below the hydrotheca^, distal extremity slightly curved to one side, the terminal aperture, large, circulai'. Length of largest specimens, 100""". San Diego, Cal., — Henry Hemphill ; taken from kelp roots washed ashore during a storm. San Diego, Cal.,— D. C. Cleveland, 1875; Santa Cruz, Cal., Bay of Monterey, — Dr. C. W. Anderson ; Vancouver Island, — J. M. Dawson. This variety is very similar to S. abietina of Linmeus in many *S'. F. Clafh — Ilydrokh of the Pacific CoaM. '251 respects. It lias the same mode of growth, the same robust habit, tlie same style and airangement of hydrothecse. There is quite a dif- erence in the natural size figures of .S. ahietlna given by Uincks and by Johnston. Our specimens from the New England coast agree very well with Johnston's figure, which is just about twi(!e the size of Hincks' H""ure. Sertularia argentea Ellis and Solander (Linn.) Three fine specimens of this widely distributed species have been found at Santa Barbara, California ; they are in good condition and loaded wdth gonothectB. The only characteristic which shows any variation is the gonotheca?, which are proportionally a trifle longer and also have the orifice a little larger than in our east coast speci- mens. This slight difference being the only one and this character varying considerably in the same specimen, I should not think of sep- arating them. Santa Barbara, Cal., — INtrs. Ellwood Cooper. Height of largest specimen, 160""". Sertularia Greenei Murray. Sertularia tricuspidata Murray, Ann. and Mag. for April, I860, p. 250, PI. XI, lig. 1. Sertularia Greenei Murray, Ann. and Mag., v, p. 504, 1860. Cotulina Greenei A. Aga.ssiz, Cat. of N. Amer. Acalepha?, 1865, p. 147. Plate XXXVIII, figure 6. Stems erect, slender, densely clustered, simple, thickly branched, basal portion straight, above the first branch becoming slightly flex- uous, forming a graceful arc between each two branches, color cor- neous, usually darkest at the base ; joints placed at right angles to the stem and very irregularly distributed, forming iuternodes which bear from one to eight pairs of hydrothecEe ; branches alternate, erect, many of them short; some of the lower ones equal in length to the main stem and closely resembling it in every respect; others from the middle portion of the stem are of a medium length aiul, like the longest branches from the lower part of the stem, reach to the extrem- ity of the main stem forming a corymb-like structure, Tlie branch- lets, like the branches, are mostly short, but a few are of considerable length, extending to the ends of the branches. Ilydrothecse sub-alter- nate, tapering uniformly to the distal end, Avith oblique, toothed apertures, which face toward the stem ; on the outer edge of the aperture are two large, prominent teeth separated by a deep notch. Trans. Conn. Acad., Vol. III. 33 June, 187G. 258 ;S'. F. Clark — Hydroicts of the Pacific Coast. Gonotheciu borne in rows on the upper sides of the branclilets ; the upper j>ortion cylindrical, the proximal half tapering toward the base, aperture terminal, in a small cylindrical process elevated from the center of the distal end. Height of largest specimens, 90""". Bay of San Francisco, Cal., — Murray ; Santa Cruz, Cal., — Dr. C. W. Anderson; Santa Barbara, Cal., — Mrs. EUwood Cooper; Vancouver Island, — J. M. Dawson. This is an interesting form as it is the only member of the Sertu- laridm on the American coast having the peculiar aperture to the goriothecae, by Agassiz called bottle-shaped, though it is by no means an uncommon form among our CainjKinidar'uloe, — Obelia yelatlnosa, 0. genicnlata and 0. dichotoma having the same general form of gonotliecae. A peculiar discrepancy occurs in the descriptions of Murray and A. Agassiz in regard to the number of teeth on the rims of the hydrothecae ; the former describes them with three teeth, the centi'al one being larger than the two lateral, while Agassiz describes them with four, two prominent exterior points and two smaller ones near the stem. We have quite a large supply of specimens in a good state of preservation and after having carefully examined them all, I cannot find a single hydrotheca that would afford any reason for changing the above description of tvio teeth upon the rim of each cell. I should judge from Murray's figure that his specimens were not well preserved and by contracting had thrown out the inner margin of the rim, giving it, in some views, the appearance of a tooth. But how he made out one tooth to be much larger than the other two, I am at a loss to understand. And the fact of Agassiz having seen four teeth I am at present unable to account for. Sertularia furcata Trask. Sertularia furcata Trask, Proc. Cal. Acad. Nat. Sci., Mar. 30, 1857, 112, Plate V, figs. 2, a, b, 0, d, e. Plate XXXIX, figure 3. Stems short, unbranched, rooted by a creeping stolon, simple, spread- ing in every direction forming dense verticillated clusters around the pieces of fucus on which it is usually found, attached to the stolon by a short, slender, twisted process about the length of an internode, divided by transverse joints into short regular internodes each bearing a single pair of hydrothecte, color corneous. Hydrothecije oppo- site, deeply immersed in the stem, with two large, sharp teeth on iS. F. Clark — Hydro ids of the Pacific Coast. 259 the outer margin and a large aperture generally reaching to the stem. Gonotlieca3 large, sessile, generally borne near the base of the steins though occasionally found scattered over the entire length, of an elongated oval form, sometimes slightly compressed, with a large circular, terminal aperture. Height of largest s|)ecimen, 45""". Bay of San Francisco and Farallone Islands, — Trask ; Santa Cruz, Cal., Bay of ^lonterey, — C. W. Anderson ; San Diego, Cal., — Dr. E. Palmer ; Santa Barbara, Cal., — Mrs. Ellwood Cooper ; Santa Barbara, Cal.,— Dr. L. X. Dimmick. S. furcata seems to be more nearly allied to aS', Greenei than to any other Sertularian of the west coast known to us, both having the same style of hydrothecte, arranged in about the same manner and with similarly toothed apertures. It is the same style seen in S. operculata of Linnteus. -iS'. furcata may be readily <listinguished from S. (xreenei by its entirely different gonothecfe, by the different size of the aperture in the hydrothccae, by the extent to which the hitler are immei-sed in the stem and by the habit or mode of growth. In general appearance it strongly reminds one of the Sertalaria puudla of the New England coast. Sertularella turgida <'iaiis (Trask). Sarlularia turgida Trask, Proc. Cal. Acad. Nat. Sei.. Mar. 30, 1857, 11."., Plate IV, fig. 1. Sertularia turgida A. Agassiz, Cat. N. Am. Acalepha3, p. 145, 1865. Plate XXXVIII, figures 4, 5. Stems attached by a creeping stolon, sparingly branched, attached to the stolon by a pedicel consisting of three or four rings, short, stout, simple, spreading in every direction from the branches of fucus and pieces of laminaria which seem to be the favorite stations of this species, divided by oblique joints into shori, stout internodes each bearing a single hydrotheca, color light corneous ; branches stout, erect, usually about half the length of the main stem and very irregu- larlv arranged, in some cases alternately, in others all the branches spring from one side of the stem and sometimes there seems to be no regular arrangement. Hydrotheca? large, full, alternate, deeply immersed in the stem, the inner angle of the proximal end extending more deeply into the stem than the outer, aperture large, armed with three stout teeth, two of which are larger than the other and are situated on the outer side of the rim, facing the stem, the third tooth 260 iS'. F. Clark — Hydroids of the Pacific Coast. is on the inner margin of the rim midway between the other two. Gonothecae, arising in the axils of the hydrotheca?, are large and of two forms ; the larger form, similar to the gonotheea of Sertu- larella polyzonias Gray (Linn.), is obovate, sessile, armed with a few stout, blunt spines around the distal end, aperture, terminal and at the outer end of a small cylindrical process formed by a constriction or a very sudden tapering near the extremity and surrounded by a number of the largest spines ; the smallei' form is supplied Avith about twice as many spines as the larger foi'm and is shorter and proportionally broader, the broadest portion being nearer to the distal end ; this variation in form undoubtedly indicates a sexual difference, the smaller form proba1>ly being the male find the larger the female gonothecje ; both forms have the surface more or less roughened by transverse Avrinkles. Length of largest specimen, 38""". Bay of San Francisco, Monterey, Tomales Point, Cal„ on mollusca and alga^, — Trask ; Santa Ci-uz, Bay of Monterey, Cal.,— Dr. C. W. Anderson ; San Diego, Cal., — D. C. Cleveland ; Vancouver Island, — J. M. Dawson. The nearest ally of S. turglda is the ^S. polyzonias of Gray, to which in many respects it bears a striking resemblance. Hydrallmania Franciscana Clark (Trask). Plumularia Franciscana Trask, Proc. Cal. Acad, of Nat, Sci., vol. i, p. 113, PI. IV, fig. 3. Plumularia gracilis Murray, Ann. and Mag. of Nat. Hist, for April, 1800, p. 251, PI. XII, fig. 1. Trask and Murray both had representatives of a species which they referred to the genus Plwnularia and whicli, from their descriptions and figures, undoul)tedly belongs to Hinck's genus Ilydmllmania, though at the time their descriptions were published, this genus had not been recognized. This sjiecies is certainly very close to II. falcata of Hincks, but Murray, who had an opportunity of comparing the two, says they are distinct. Unfortunately we have had no specimens. Bay of San Francisco, Cal., among rejectamenta on the beach, —Trask. >S. F. Clark — Hj/droids of the Pacific Coast, 261 Plumularia setacea Lamarck. Sertularia piiinata, [3, Linn., Syst. Nat., 11! 12. Sertularia setacea Pallas, Elench., p. 148. Plumularia setacea Lamk.. An. s. Vert. (2d ed.), ii, 165. Plate XLI, figures 1, 2. Stems simple, slender, erect, rooted by a creeping stolon and divided by transverse joints into short internodes of uniform size, regularly branched ; pinn.e alternate, regulai-ly arranged, one from each internode, arising from the stem by a pi-ominent process pro- duced from the outer and upper side of each internode, divided by trans- verse or slightly oblique joints into internodes of two sizes arranged alternately. In large specimens 150"'"'. to 800'""'. long, the main stems are considerably branched, the branches alternately arranged and clustered, extending quite or nearly to the distal end of the stem; the branches give off" l)ranchlets, which like themselves resemble the main stems in every particular. Hydrotheca? with an even rim, small, borne on the larger sized internodes of the pinna>. Nematophores compound, those on the pinn;v not quite equal in length to the hydro- thecfe, those on the main stems a little longer than the hydrothecae; three on each internode of the stem, two in the axil of each pinna, one on the opposite side of the internode near the bnse, one only on the upper side of the smaller joints of the pinn{\3 and three on the- larger joints, one just below the hydrotheca, and one on each side of the apertui-e Gonotheca^ sessile, l)orne in the axils of the pinna* ; female elongate oval, produced at the distal end into a tubular neck with a discoidal, terminal orifice ; male, smaller than the female, fusiform, and with a much smaller aperture. Santa Cruz, Cal., — C. W. Anderson; San Diego, Cal, — Dr. H Pal- mer; San Diego, Cal., — H. Hemphill; Vancouver Island, — Dawson. Most of our specimens consist of dense clusters of the delicate shoots, about 50'""'' to 80"""* long, and usually attached to some large, coarse alga. Ellis' old name of " iSea Bristle.^'''' was well chosen, for it conveys quite an accurate idea of the appearance of these smaller forms. The larger forms are more branched, usually of a darker color and have a closer resemblance to hair than to bristles; one of our largest specimens from San Diego consists of a tuft 200"""- in length composed of about three hundred branched shoots ; this liad been washed ashore and was found by Dr. Palmer; a still larger specimen was dredged off San Diego in six to ten fathoms by IVIr. Hemphill, which measures .300"""- in length and forms a thick cluster of about a thousand shoots. 262 *S'. F. Clark — Uydrolds of the Pacific Coast. Aglaophenia struthionides Clark (Murray). Flumulari'i stridhmmles Murray. Ann. and Mag. of Nat. Hist, for April, 1870, 251, Plate XII, fig. 2. Arjlanphenia franciscana A. Agassiz, Cat. N. A. Acalephai, p. 140, 1865. Plate XLI, figures 3-3''. Stems rooted by a creeping stolon, simple, erect or spreading in every direction, divided by slightly oblique joints into very short internodes of equal length, each bearing a single pinna, varying from the lightest to the darkest horn-color; shoots tall, stout, plumose, tapering slightly toward the base, the distal end abruptly pointed ; pinna' slightly curved, sub-erect, unbranched, not in the same plane, the sides bearing the hydrothecie curving toward each other, divided into short internodes by slightly oblique joints, each internode bear- ing a single hydrotheca. Hydrotheca^ large, cu}>shaped, expanding toward the distal end, aperture large, patulous, rim denticulated, armed usually with eleven, sometimes nine, sharp, uneven teeth. Nematophores tubular, the lateral ones of medium size, projecting ear-like from the sides of the hydrothecae, the anterior one long, adnate for the greater part of its length, free near the distal end, extending nearly or quite to the edge of the toothed rim, aperture small, discoidal, terminal ; those upon the corbula3 are a trifle larger than the lateral ones and are arranged in transverse rows, the ends of which do not meet. Corbuhe large, cylindrical, with numerous rido-es (ten to sixteen) composed of oblique rows of nematophores ; usually from two to six hydrotheca? at the base of the corbula. In luxurious specimens the corbula- are very abundant, there being between seventy and eighty t)n a single shoot. Length of largest specimen, 150"""" Bay of San Francisco,- -Trask ; San Francisco, Cal.,— A. Agassiz; Santa Barbara, Cal.,— L. F. Dimmick ; Santa Barbara, Cal.,— Mrs. EUwood Cooper ; Santa Cruz, Cal., Bay of Monterey, — C. W. Ander- son* San Diego, Cal.,— D. C. Cleveland; Vancouver Ishmd, — Miss Mitchell ; Vancouver Island, — J. M. Dawson. In the various lots of Hydroids which we have received from the western coast, this species has always been the most abundant. It seems to be as common and as widely disti'ibuted on the western coast of the United States as tSertularia puniila is upon the eastern coast. Both are also very often found parasitic on algiie, but A. stru- thionide-s is often found in deeper water than N. pui/td-a. S. F. Clark— iTydroich of flic Pacific Coast. 263 As ^Murray r(>inarl<s, tliis species is nearly allied to Phinmhirla rrf'sfafa, the A. phmui of Liuna'us, but is imicli elosev in lialtit, tiie liydrotliecfB are Avider-inouthed and shallower and the teeth upon tlie rim are unequal. An inipoi-tant error occurs in the synonyniv ol'liiis species given bj'^ Mr. A. Agassiz. lie has, under the name .\<ihio- phenia Franciscana the foHowing synonymy : Plumularai Franci.'^caiia Trask. Plumularla struthiontdes Murray. The Pluriiidarla Franciscana of Trask l)elongs to tlie geiuis Jfi/- drallmanki: of llincks, as a glance at the figures and description of Trask will show; and it is synonymous witli the Phnnalarid gracilis of Murray. Hence the name of the above descriljed species should not be A. Franciscana, but .1. strut hionides. EXPLANATION OF PLATES. ' Plate XXXVIII. Figure 1. — Eudendrium, s]). ; from Santa Cruz, Cal. Figure 2. — Tubularia elegans; a, cluster of medusas buds; a, a', and«", buds in differ- ent stages of development ; a'", an actinula escaping. Figure 3. — Bimeria gracilis ; a, a' and a", sporosacs. Figure 4. — Sertularella turgida ; a, the gonangium or gonotheea. Figure 5. — Sertularella tur/jida ; another form of gonotheea. Figure 6. — Sertularia Greenei ; a. and a', the gonangia (gouothecte) ; h, hydrotheca. Plate XXXIX. Figure 1. — Campanularia cylindrica ; la, the same showing the full length of the pedicel; 1 fc, the female gonotheea ; Ic, an abnormally dev^eloped 'lydrotheca ; \d, the male gonotheea. Figure 2. — Campanularia Jusiformis ; '2a, '2h, and 2d, the same showing the amount of variation in the width and depth of the hydrotheca? ; 2e, 2/1 and 2(/. the gonothecse ; r, the rootstock or creeping stem. Figure 3. — Sertularia furcata ; a, and a', the gonothecte. Figure 4. — Campamdaria everta; a, and a', the gonotheea?; r, the creeping stem. Figure 5. — H^lecium tenellum ; from San Diego, Cal. ; r, the creeping stem. Plate XL. Figure 1. — Sertularia anguina; la, a single hydrotheca showing the outline of the outer margin of the rim. Figure 2. — Sertularia anguina; a portion of the main stem. Figure 3. — Sertularia anguina, var. rohusta; a portion of the main stem. Figure 4. — The same ; portion of a branch ; a, gonotheea. Figure 5. — The same; with a monstrosity, a. at the extremity of the branch. 264 ^. J^l Clark — Hydroids of the Pacific Coast. Plate XLI. Figure 1. — Plumularia setacea ; portion of a branch with neinatophores and female gonothecaj, a. Figure 2. — The same; a portion of the main stem; n, uematophores ; /;, hydrothecaj ; 2a, male gonotheca. Figure 3. — Aglaophenia struthionides ; a portion of a pinna ; 3a, 36, and 3c, different views of the same; n and «', nematophores ; h, hydrotheca; 3tZ, cortaula; «, the body of the eorbula; 6, the wing-like processes at the base; n, and -«', the nematophores; /*, the hydrothecte. VII. On the Anatomy and Habits of Nereis virens. By Frederick M. Turnbull. Read January 19, 1876.* The Nereis [Alitta) virens^ which is one of the Largest and most common of our marine annellids, is found under stones or burrowing in the sand and mud of sheltered shores, both at low-water mark, and at a considerable distance farther up. It grows to the length of eighteen inches or more, and is quite stout in its proportions. It is very active and voracious, feeding on other worms and vari- ous kinds of marine animals which it finds when burrowing in the sand.f It will even devour its own immediate relatives, if hungry when it meets them. It suddenly thrusts out its proboscis and seizes its prey with the two powerful jaws, then withdraws the pro- boscis, the jaws closing at the same time. In this way it w'ill tear large pieces from the body of its victim, being able, at one bite, to cut in two a worm of its own size. One which I had confined in a small dish of water, bit its own body in two pieces at the middle. As the proboscis is turned inside out, when it is pro- truded, whatever has been siez<'d by the jaws will be drawn by them inside the proboscis as soon as the latter resumes its natural jjosition, the proboscis then acting as a sort of gizzard. These worms, by secreting a viscid fluid, will surround themselves in a few minutes with a translucent sheath which binds the grains of sand together, forming a loose and flexible tube. They remain most of the time in these tubes, Avhich are nearly always situated in sand and mud or under flat stones, and they move in them with consider- * Abstract of a graduation thesis presented at tlie Sheffield Scientific School, June, 1875. f Later observations show that this species does not restrict itself to an animal diet. Several large specimens, taken by me in October, 1875, had the intestine com- pletely filled witli algte of several species, among which Ulva latissima was most abundant. The algas were torn into fragments and large shreds and rolled together into long pellets, but even after passing through the intestine their nature could be easily recognized. — a. k. verrill. Trans. Conn. Ac.\d., Vol. III. 34 August, 1876. 266 F. M. Turnlmll — Aviatomy and Habits of JSfereis virens. al)le freedom and rapidity, pusliiiig themselves along by means of their aciculse, setse and ligula^, the acicuhe being controlled by special muscles. They hold their tentacular cirri straight out in front of them, as they move, in order to have warning of anything that they may approach. The tautog, scnp and other fishes dig them out of the sand and devour them eagerly. But at certain times, especially at night, they leave their burrows, and swim about like eels or snakes, in large num- bers, and at such times fall an easy prey to many kinds of fishes. This habit seems to be connected with the season of reproduction. They were thus observed swimming at the surface in the day time, near Newport, in April, 1 872, by Mr. T. M. Prudden and Mr. T. H. Russell, and several times by Professor Verrill, later in the season. At Watch Hill, R. I., April 12th, 1873, Professor Verrill found great niimbers of the males swimming in the pools among the rocks at low- water, and discharging their milt. The males were also seen swim- ming in the tide-pools and shallow waters at Savin Rock, April, 1875, by Professor D. C Eaton and Mr. Kleeberger. The JVereis virens is abundant at all seasons of the year, in most places along the sandy and muddy shores, both of the sounds and estuaries, burrowing near low-water mark. It occurs all along the coast, from New York to the Arctic Ocean, and is also found on the northern coasts of Europe.* The body consists of a large number of rings or segments. This number varies with the size and age of the worm. It may be less than one hundred, or as many as two hundred. The increase in length takes place by the addition of new posterior segments, in advance of the caudal segment. New segments may also be formed when a part of the body is broken oif, and in this way a considerable part of the posterior portion of the body can be completely reproduced. The head (figs. 1 and 17) is very fully developed, being provided with two pairs of eyes and two pairs of antennae. It is attached to the dorsal side of the first segment, wliich is called the buccal or mouth-ring (figs. 1 and 15, d), because it contains the mouth (fig. 15, ni). There are also four feelers, called tentacular cini (fig. 1, ee, ee', e and e'), arising from the buccal ring on each side of the head. The abdominal rings (fig. 1, g) follow the buccal ring. Each one of these has a pair of lateral lamelliform appendages (figs. 1 and 1 7, A), used as paddles in swimming and also Serving the purpose of gills. The last segment or ring of the body bears a pair of cirri, similar to the tentacular cirri, and also contains the anal orifice. * See, for habits of this and allied species, Professor VerriU's report in First Report of U. S. Commission of Pish and Fisheries, p. 318. 1873. F. M. Turnhidl — Anatomy and Hahits of Nereis virens. 267 The mules are easily distinguished from the females by ditferences in color and in the form of the side appendages. The color of the body in the male is an intense steel-blue, which blends into green at the base of the lateral appendages. These have a rich and brilliant sea-green color which is heightened by the complimentary effect of the luimerous red blood-vessels they contain. The latter are especially noticeable toward the posterior end of the body where the skin is thinner and less opaque, so that the appendages, with their network of ca|»illaries, appear bright red. In the female the body is of a dull greenish color, with a slight tinge of orange and red. The appendages are orange-green at the base, and become bright orange toward their extremities ; but sometimes they are greenish throughout. The whole surface of the skin, in both male and female, is iridescent, reflecting bright hues when placed in the light. The head (figs. 1 and 15, a) is small, and flattened on the doisal and ventral sides. From the position of the anterior eyes the sides taper toward the anterior extremity, where it is rounded oft' and terminated by a pair of small antenna (figs. 1 and 15, h, b). There are two pairs of eyes on the upper surface of the head, one pair near its base and another pair more anterior and farther apart. The anterior eyes are situated near the middle and on the broadest portion of the head. On each side of the head, attached to its anterior half and also to the buccal ring, is a large antenna or palpus, as it is sometimes called (figs. 1 and 15, c). These are stout, fleshy and somewhat contractile organs. Each has a small rounded lobe at the tip (figs. 1 and 15, c'). The buccal segment and the head constitute the cephalic or head region. The tegument about the mouth is wrinkled and folded lon- gitudinally, presenting an appearance like the mouth of a purse, when drawn together by strings. The tentacular cirri are long, slender and quite flexible. They receive their nerves from the first abdominal nerve-ganglion, whereas the antennae receive theirs from the head- ganglion. There are four pairs of tentacular cirri, two dorsal (fig. 1 , ee, ee') and two ventral (fig. 1, e, e'). The ventral ones are situated nearer to the palpi than the doi'sal, and hence the two are called respectively the internal and external tentacular cirri. The relative lengths of the ten- tacular cirri are shown in fig. 1. The two large dorsal tentacular cirri (ee, ee) are longer in the male than in the female. In the male they will reach to about the middle of the ninth segment, when laid along the back. In the female they will reach to about the middle of the 268 F. M. Turnhull — Anatomy and Habits of Nereis virens. fifth segment. The other tentacular cirri have more nearly the same length in both sexes. The abdominal segments increase in size to about the eighth, and then remain nearly the same for some distance along the middle region, but the posterior rings decrease in size, causing the body to taper until it becomes quite slender. The appendages, also, are longer and broader along the middle region than toward either extremity. On the ventral side of the segments the part continuous with the feet is smooth, l)ut the other parts show the strong transverse muscles (fig. 26), which, by their contraction, lessen the size of the body cavity. The lateral appendages or feet of Nereis virens are quite compli- cated and wonderful organs. They are biramous (fig, 12), having two rami, one dorsal (A) and the other ventral (B). The aciculse of each foot arise from a crypt which is attached by shroud-like muscles to the base of the foot. The sette arise from the interior of the two rami. The feet are complicated by the addition of other organs, serving for locomotion, sensation and respiration. The respiratory organs, often called liguhne, are moi-e or less flatten- ed lobes with their teguments very thin and filled \\\i\\ a rich vascu- lar netw'ork (figs. 22, s^ 25, 26). The upper ramus has two ligulae, a superior one (fig. 12, h) on its upper, and an inferior one (fig. 12, d) on its lower side. At the base of the superior ligula, on a sort of shoulder of its upper edge, is the dorsal cirrus (fig. 12, a). There are also two setigerous lobes smaller than the liguhe, one on each side of the open- ing through which the sette protrude (fig. 12, c and k). The anterior (c) is longer than the other [k) and is connected with the inferior ligula {d) of the upper ramus. The acicula is attached to the inner walls of the ramus forming a partition, which terminates with the end of the acicula between the lolies c and d, (fig. 12) and generally forms, in the middle and posterior parts of the body, a third and smaller lobe (fig. 12, r'). The lower i-anius lacks a superior ligula, but it has an inferior one (fig. 12,^) more rounded and not so broad as the others. In the lower ramus, as in the upper, there is normally only one fascicle of setae, but in the lower ramus it is divided into two clusters by the acicula, which, by its attachment to the inner walls of the ramus, forms a par- tition. Here, as in the u])per ramus, there are two flattened setiger- ous lobes, about equal in length, one on each side of the o])ening for the setae (fig. 12, / and e), and the partition formed by the acicula extends to the extremity of the anterior one, dividing it into two small lobes (fig. 12, e and e). p. M. Turiihidl — Anatomy (otd Ilahits of Nereis vlrois. '269 The inferior ligiihi of the u])per nuiius, with its two hjhes (tig. 12, c, d) corresponds, apparently to the lobe (e) and its divisions, while the lobe {k) of the upper ramus corresponds to the lobe (/) of the lower ramus. In the upper ramus, the acicula and its partition do not divide the bunch of setae, as it coAies out above the acicula. The ventral cirrus arises from a slight protuberance of the inferior side of the lower ramus (fig. 12, h). The feet are not all alike from one end of the body to the other ; they change in form most along the anterior region, and in the first five the variation is considerable ; along the middle and posterior regions, it is slight in comparison, I have figured tlu' first five, the forty-fifth, and the one hundred-and- ninth feet of a female worm, having one hundred and twenty-one segments; also the first, fifth, forty-fifth, and one hundred-and-ninth in a male worm of nearly equal size, but having one hundred and sixty- one segments. These figures show the posterior view of the feet. Those of the female will be first described, and then compared with those of the male. In the first foot of the female (fig. 3) the upper ramus has only the doi'sal cirrus and the superior ligula developed. The former is about one-fifth longer and a little less than one-third as broad as the latter, being very well dcA^eloped, wdiile the ligula is somewhat rudimentary. The ligula is rounded and simple in shape, having a constriction at its base. Of the lower ramus, all the parts are present. The posterior setigerous lobe (/') is longer than the anterior setigerous lobe (e), and is oval and flattened. The anterior setigerous lobe {e) is seen to be divided by the acicula into two smaller terminal lobes (e and e'), in- dicated by dotted lines where they are covered by the posterior seti- gerous lobe (/"). The inferior ligula {g), of the lower ramus, is of about the same size and shape as the superior ligula [h], and the infe- rior cirrus (//), is like the superior cirrus {<(). The setse of the lower ramus, as shown in the figure, extend just Ijeyond the posterior seti- gerous lobe (/'). They are somewhat rudimentary and are for the most part blunt. The acicula is also rather small. The second foot (fig. 4) has the anterior setigerous lobe (e), and the acicula a little larger and better developed, and the dorsal cirrus is seen to be a little shorter than that of the first foot; otherwise the second foot is scarcely different from the first. In the third foot (fig. 5) the upper ramus has developed an inferior ligula, setjB and acicula. It also has an anterior setigerous lobe (c). This lobe is a part of the inferior ligula [d). It shows more 210 F. M. Tumhidl — Anatomy and Habits of Nereis virens. distinctly in some of my figures of the feet of the male. The superior ligula is of about the same size and shape as the inferior ligula. The lower ramus is a little more developed than it is in either the first or second feet, but the inferior cirrus (A) is shorter. The fourth foot (fig. 6) is more highly developed than the third ; but the inferior cirrus is shorter in comparison with the lower ramus. The fifth foot (fig. 7) is still more highly developed than the fourth, and the superior ligula is larger than it is in the preceding feet. The dorsal and ventral cirri are both smaller. If we now compare the fig- ure of the fifth with the figures of the first and second feet, the differ- ence is seen to be considerable, particularly in the length of the cirri. The superior ligula {b) begins to increase in size at the fifth foot, and continues to do so, until in the forty fifth (fig. 8, h) it is larger than any other part of the foot. It has also become flat and pointed. The shoulder (?") is much larger. The inferior ligula [d) is also flat and pointed. A posterior setigerous lobe {k) is now large enough to be easily seen. The posterior setigerous lobe (/) of the lower ramus is somewhat pointed, and its lower edge is oblique. The anterior setigerous lobe (e) is now as long as the posterior seti- gerous lobe, and its two divisions are nearly equally developed. The inferior ligula {g) is rounded and somewhat tapering toward its extremity. The dorsal and venti'al cirri are now quite small, particu- larly the latter, while otherwise the foot is much larger. The setse have become longer and more perfect, from the anterior toward the posterior, attaining their maximum in the middle region of the body. The aciculiB are also larger here than in either the anterior or pos- terior regions ; although, in the posterior region they are longer in comparison with the size of the foot than anywhere else. The one hundred and ninth foot (fig. 9), as shown by the figures, is nearly the same in form as the forty-fifth, but smaller. On comparing the first, fifth, forty-fifth and hundred and ninth feet of the female with the same in the male, we find that there is a marked difference in the cirri. The dorsal cirrus in the first foot of the male (fig. 10) is one-half longer than it is in the female, and it is larger in proportion. In the fifth foot (fig. 1 1 ) the difference is the same. In the forty-fifth (fig. 12) the dorsal cirrus of the male is longer and also slenderer than it is in the female. In the one hundred and ninth foot (fig. 13) the dorsal cirri are about the same in length, but those of the male are more slender. The difference between the dorsal cim decreases toward the posterior end of the bodi/, being greatest in the anterior segments. F. M. Turnhnll — Anatomy and Habits of ]Vereis virens. 271 The ventral cirrus of the first foot of the raale (tig, 10) is not quite so long as the dorsal cirrus, but it is a little longer than the venti'al cirrus of the female. In the fifth foot (fig. 11) it is a little longer and much more slender than it is in the female. In the forty-fifth (fig. 12) it is twice as long and of the same diameter as it is in the female. In the one hundred and ninth (fig. 13) there is the same difference which we see in the foi'ty-fifth. The difference between the ventral cirri of the male and female, in- creases from anterior to ^yosterior, being greatest m the posterior seg- ments. Beginning with the forty-fifth foot (fig. 12) a shoulder {x) appears on the dorsal side of the lower ramus in the male. It increases in size and definiteness farther back, but finally disappears. This shoulder is never seen in the female, and is a good character for distinguishing the sexes, but it requires microscopic examination. The difterence between the cirri of the male and female is suflicient to distinguish them without the use of a lens. The setae under the microscope are very delicate and beautiful (figs. 2. la). They consist of two parts, the shaft (a) and the blade (J). The shaft has a transversely striated appearance, which is exceedingly regular. The blade is held in a sort of socket in the end of the shaft, and one edge is toothed like a saw. There are two forms : one in which the blade is short, having its extremity blunt and slightly hooked (fig. 2), and one with the blade tapering to a fine point, the blade being long and slender (fig. 2a). The length varies and with it the delicacy of the point. In most of the setae the latter is so sharp that it seems to vanish, and can be seen only with a high power. Those setse which are hooked have the hooks and toothed edge turned upward; and these are always confined to the lower ramus, and to the lower side of the latter in both bunches. In the middle and posterior regions these hooked setae disappear, their place being taken by the other kind ; but the latter are shorter than those in the upper part of the bunch. The same forms of seta? are found in both male and female. Those of the anterior feet are shorter than those in the middle region. When viewed by transmitted light the bunches present all the colors of the spectrum. The aciculge (figs. 12 and 22) are simple thorns, in the form of an elongated cone, generally a little bent. They are black, except at the base, showing through the translucent integuments of the foot. At the base they are hollow and therefore lighter colored. 272 F. M. Turnhull — Anatomy and Habits of JSTereis virens. Muscular System. The walls of the body contain two muscular layers, which are well defined. The first is thick on the ventral side of the body (fig. 22, k), but is thinner elsewhere. It forms in each ring a muscular plane of transverse fibres. The second is placed within the jjreceding and is formed of great fascia? which are attached to the interannuiar partitions (fig. 16, b). This layer does not exist on the ventral side, between the bases of the feet. The interannuiar partitions (fig. 16, a) are attached to the intestine, which they hold in place, dividing the general cavity of the body into a series of chambers (fig. 16) ; but as the inner subcutaneous layer does not extend over the ventral floor of the cavity, the chambers open into one another beneath the intes- tine. I have already mentioned the muscles attached to the fleshy knobs, which hold the aciculfe. These muscles (fig. 22), when they contract all together, force the aciculae outward. When difterent ones contract they move the aciculffi to one side. The Nereis virens in swimming moves its body laterally, like a snake. It sometimes has also an undulatory movement, up and down. These motions are all produced by the subcutaneous miiscles. The lateral appendages are used as paddles, but their principal use is to push the worm along in its tube, and for crawling. They are greatly aided by the stifle aciculffi, controlled by their special muscles (fig. 22), and by the seta\ When burrowing, the proboscis is used to push away the sand in front and is then withdrawn, while the body is moved forward partly by a vermicular motion and partly by the side ap- pendages. Alimentary Syste)n. The proboscis is a very remarkable organ and constitutes a formid- able weapon. It is divided by M. DeQuatrefages into three regions: the pharyngeal, the dental, and the oesophageal (figs. 16-18). The mouth has already been partially described. The walls of its cavity are connected by several small, delicate muscles, with the walls of the body cavity, as I have shown in figure 16, m and n. The pharyngeal region commences immediately back of the buccal cavity (figs. 16 and 20, x), and has two muscular partitions (fig. 16, c). The dental region (20 and 16, jo) is very muscular, and is provided with a considerable number of small teeth, or denticles, which are arranged in groups on the anterior, inner surface of its walls. It also has two large and powerful jaws attached by their hollow bases to the muscles of the posterior inner surface. The worm has the power of turning this dental region inside out. F. M. Turnhull — Anatomy and Habits of Nereis virens;. 273 Whoii the proboscis is iiisido the body, it takes the position sliown in tiniires 16 and 20, tlie cesojthagus being curved and pushed back against the intestine ; but when it is forced out, the oesophagus straightens, and that part of the proboscis which is protruded takes the position shown in figures 1 7 and 1 8, these figures sliowing the ar- rangement of the jaws and denticles, figure 17 giving the dorsal view and figure 18 the ventral view; the buccal ring is very much stretched ; and the head, antenna^ and tentacular cirri are forced over on the back (fig. 17). The jaws are imbedded in and attached to the special muscles, in such a manner that the more the proboscis is l^rotrudcd the farther apart their points move, and when the probos- cis is withdrawn they close like a pair of scissors, their points crossing. The jaws (fig. 19) are curved like hooks, and have their inner concave edge denticulated with about ten teeth. They and the denticles are composed of a black chitinous material. The denticles, which are conical and pointed, are not attached to muscles, but are simply im- bedded in the surface. The number and size- of the denticles, and even their positions, vary considerably in the different specimens ex- amined. T think it would be hard to fi.nd two specimens exactly alike in this respect. Among seven worms, I found five with one denticle, one with two denticles, and one with seven denticles on the median anterior area of the dorsal side (fig. 17, o). Among six worms, I found two with two denticles and four with none at all on the posterior median area of the dorsal side (fig. 17, t). On the left submedian anterior area of the dorsal side (fig. 17, I), the number of denticles varied from three to eight ; and on the right sub- median anterior area of the dorsal side (fi:g. 17, I') from two to eight. In only one case did the two last areas have the same number of den- ticles. On the left submedian posterior area of the dorsal side (fig. 17, s) the number of denticles varied from none to five; and on the light submedian posterior area of the dorsal side (fig. 1 7, s') from one to four. The denticles on the lateral and ventral posterior areas (fig. 17 and 18, r) vai-y considerably in number, position and size. Among four worms, the number of denticles on the right lateral anterior area (figs. 17 and 18, n) varied from twelve to twenty-five; and on the left lat- eral anterior area (figs. 17 and 18, n') from eleven to thirty. Among five worms the number of denticles on the anterior median area of the ventral side (fig. 18, y) varied from four to twelve. On each of the two submedian * areas of the ventral side (fig. 18, x') there was one denticle. Trans. Conn. Acad., Vol. III. 35 August, 1876. 274 F. 31. Tarnhull— Anatomy and Habits of Nereis virens. In some of the large worms the denticles are as small as those of smaller worms, while in other cases they are much larger. The large denticles probahly are knocked off in some way and new ones grow in their place. This would account in part for the large number of very small denticles, and also for the variations in number. Attached to the anterior end of the oesophagus, one on each side, are two salivary glands (figs. IH and 20, j). These are free except at one end, and are ciliated on their outer surface. The intestine proper (figs. 16 and 20, r) is straight and is constricted somewhat by the muscular partitions of each segment through which it passes. It is brilliant greenish yellow in color and is surrounded by a regular cap- illary network of blood vessels (fig. 20). The internal surface of the oesophagus is tessellated with low, rounded papillae or tubercles, "^rhese are regular in shape and equal in size. Their sides are diag<mal to the length of the oesophagus. They show through the walls of the oesophagus, so that its outside appears tessel- lated with dark squares, and as the oesophagus is stretched or con- tracted they become diamond shaped. The tubercles are of a dull color, between brown and yellowish green. The intei'ior surface of the intestine is also covered with regular longitudinal rows of low, rounded tubercles, which are much smaller than those of the (esophagus. They are greenish-yellow like the out- side of the intestine. The end of the oesophagus projects into the cavity of the intestine, and its opening, which has sphincter and also longitudinal muscles, can be enlarged or contracted to a considerable extent. The outer surface of this end of the oesophagus is continuous with and like the internal surface of the intestine. The latter secretes a brown fluid in its interior and probably acts as a hepatic organ. The dental portion of tlie proboscis acts like a gizzard, and the oesophagus is pi'obably a sort of stomach. Circulation and Respiration. The circulatory system is highly developed and complicated. The blood is red, and the vascular system is complete and closed. The principal vessels have a longitudinal course, occupying the whole length of the median line of the l)ody, one as a dorsal (figs. 20- 24, a), and the other as a ventral vessel (Ji). They are contractile, and by a sort of peristaltic motion the blood is pushed forward in the dorsal vessel, and in the op])osite direction by the ventral vessel. The dorsal vessel is visible for nearly its whole length thi'ough the more or less transparent walls of the body, and its blood can be seen F. M. 7)(r/)bi«U — Anotomy <in<] Ilahits of N'erels virens. 275 moving in a series of waves toward the liead. The ventral vessel sends oif, in each segment of the body, except a few in the region of the proboscis, two smaller vessels, one on each side. These two ves- sels fork, each sending a branch to the inferior ramus of the foot of the next segment to the rear (tigs. 20-24, /"), and another larger branch ic) around the intestine, by the side of the transverse parti- tion, to the dorsal vessel, receiving, also, on its vvay, a vessel from the upper ramus of the foot of its own segment {d). Jiesides these principal lateral vessels, there are five other vessels on each side in each segment, coming from the ventral vessel (iig. 20). These form a loose but regular network that surrounds the intestine and is connected with live other convoluted vessels, which join the dorsal vessel. This network on the intestine probably supplies the hepatic organ with material for its secretion, and very likely may receive nutritive material from the digested food. The blood moves in waves, at regular intervals, through the peripheral vessels (figs. 20- 24, c) to the dorsal vessel, but I could not see in which direction the blood moved in the network. The blood is forced into it at each pul- sation of the dorsal vessel, l)ut the normal flow may be in the opposite direction. The peiipheral vessels are also connected with this network (tig. 20). The dorsal and ventral vessels are connected at the posterior extremity of the body by a simple peripheral vascular ring (fig. 23, c), in which the blood flows from the ventral to the dorsal vessel. In the region of the proboscis, the ventral vessel sends lateral branches directly to all the feet but the first three (figs. 20 and 21). It then sends a pair of vessels to the oesophagus (figs. 20 and 21, e), which pass back along the oesophagus, one on each side, as far as the intestine, being connected with smaller vessels on the surface of the oesophagus. The ventral vessel next sends otf a pair of vessels which expand into capillary networks, one on each side (figs. 20, 21, s and g). Each of these networks sends small branches to the first three feet on its own side, and then merges into a vessel (figs. 20 and 21, A), which goes to the base of the tentacular cirri. The ventral vessel now goes upward to the under surface of the proboscis, and there divides into three branches (figs. 20, 21, t, t and /). The middle branch (fig. 21, 1) passes under a muscle and along the median line of the ventral sur- face, as tar as the pharynx, where it divides into two, forming a small vascular ring (fig. 21, n) about the latter. The two lateral branches pass upward and backward on the proboscis, each expanding into a remarkably rich and delicate network on its own side of the proboscis (figs. 20 and 21 , u). From each of these networks a vessel 276 F. M. Turnbull — .hiatomy and Mabits of Nereis vireus. (?;) passes to the base of the head, where it joins tlie dorsal vessel («), thus completing the circulation. From tliis junction small vessels probably go into the head and antennae. The lateral vessel passing to the lower ramus divides into branches ramifying on that portion which is continuous with the foot on the ventral side of the segment, and also over the lower ramus (tigs. 22 and 26). There a connection is made with vessels of the upper ramus, and I think this is done by the vessel marked x in figures 22 and 24, because it is quite large at the base of the inferior ligula of the upper ramus, and grows smaller at first and then swells out again before joining the vessels of the iipper ramus, in the superior ligula. The branch d (figs. 20, 22 and 24), coming from the dorsal ramus, receives blood from that organ and also from a peculiar and beautiful arrangement of capillaries on the dorsal side of the body (fig. 25). In the first four segments, in the region of the proboscis, the dorsal vessel has no branches, but in the remaining segments, commencing with the fifth, it has five pairs of long peripheral branches (fig, 20, e, c'), corresponding to the peripheral vessels of the posterior part of the body. They are not attached to the proboscis, but are simply con- nected with the dorsal and ventral vessels by their ends. The one coming from the dorsal vessel in the fifth segment is connected with the ventral vessel in the fourth segment (fig. 20.) The first three feet probably do not act as gills, as very little blood is sent to them. The two networks (figs. 20 and 21, u) on the proboscis are probably for carrying on the exchange between the blood and the liquid of the body cavity. The respiration is carried on by the red fluid in the beautiful ar- rangement of capillaries on the body and feet, especially the latter. The flat ligulie of the feet are exceedingly delicate in structure and take the place of gills, absorbing the oxygen from the water to purify the blood received from the ventral vessel, which then returns to the dorsal vessel. The disposition of this respiratory arrangement is shown in figures 22, 24, 25 and 26. Tlie Nervous System. ^ If Tuw - The nervous system of Nereis vi7^ens {Ggs. 27 and 28) is complicated and well developed, being composed of a series of ganglia, sending out branches and connected by nervous cords. It lies mainly on the ventral floor of the body beneath the large ventral vessel. The first and largest ganglion (figs, 27 and 28, a), analogous to the brain of higher animals, is situated in the head. It is composed of several KM. TvrnhnU — Anatomy and Habits of Nereis vlrens. 277 smaller ganglia joined together. It bears tlie eyes, on fo\ir short ner- vous peduncles, on its dorsal side. In front it sends four nerves to the antennje (/>, b and c, c, iigs. 27 and 28). Laterally it sends out two branches called the connectives [d, d), which pass around the mouth and proboscis to join the first of the abdominal gan- glia (A, tig. 27). Near the junction of the connective with the head ganglion, is a small ganglion sending nerves to the internal tentacular cirri (e, e, figs. 27 and 28). The connectives, near their lower extremity, send two nerves {g, g, fig. 27) to a series of ganglia and nerves on the ventral side of the proboscis {to, w, fig. 27). There is also an accessory connective (figs. 27 and 28, d') on each side, })ass- ing from the first abdominal ganglion to the ganglion supplying the external tentacular cirri {e',e', figs. 27 and 28). This accessory connec- tive also has a ganglion {ii\ fig. 27) at the middle, sending nerves to the muscular partitions of the proboscis. Each of the first three abdominal ganglia sends, from its anterior por- tion, on each side, a nerve that forks, one branch (fig. 27, n) going to the muscular partition and the other (o) passing through the partition to the preceding segment. In the remaining abdominal ganglia, begin- ning with the fourth, the branches ti and o become separate nerves (fig. 27). From the posterior portion of these ganglia a nervous trunk on each side (w), goes to each foot, where there is a small ganglion (k) sending off a cutaneous branch and a branch (^), supplying nerves to the foot. The ganglia (fig. 28, e, e) and the head-ganglion {a) send some very slender nerves (fig. 28, 2, z) to a series of ganglia on the dorsal side of the proboscis (y, y). In figure 27 the series of ganglia {ii\ w) are drawn as if the proboscis had been revolved about a line drawn through its anterior end, so that the ventral surface would be upper- most. In figure 28 the ganglia (y, y) are in their natural position. The dorsal ganglia (fig. 28, x, x) are connected with the ventral ganglia (fig. 27, v, v) by means of nervous cords; the dorsal ganglia (s, s, fig. 28) with the ventral ganglia (fig. 27, 1, 1), by means of nerves [)assing around the proboscis outside the points of the retracted jaws (/,/); and the dorsal ganglia {t, t, fig. 28) with the ventral ganglia (^5 p, fig.27) by means of two short, thick nervous commissures which send off the nerves {k, u, figs. 27 and 28). These two nerves {((, u) terminate in the ganglia (r, r, figs. 27 and 28). These gangba and iierves of the proboscis lie on its walls, under- neath the muscles. 278 F. M. Turnhull — Anatomy and Habits of Nereis virens. The sense-organs are the four eyes, the four antennte, the tentacular cirri, and the dorsal and ventral cirri of tlie feet ; also the long slender cirri of the posterior extremity. The antenna- and cirri are organs of touch. Organs of Reproduction. The sexes are separate, and the genital organs appear as simple glandular bodies, ovaries or spermaries, which project from the ven- tral sLiiface into the cavity of the body, between the transverse mus- cular partitions. At the sexual period they are tilled with eggs or spermatic particles, although at other times they can scarcely be seen. Neither the spermaries nor the ovaries have special excretory ducts, which open upon the surface of the body. The sperm and ova are discharged into the cavity of the body, wdiich at this period is often filled with them. At the base of the lower ramus of each foot (fig. 22, g) is a glandular body, called the segmental organs. Some of these are normally kidneys, as ui-ea has been found in them, but some are usxxally modified to act as oviducts, having a trumpet-shaped mouth opening into the body cavity and communicating with the exterior. I found the segmental organs all along the body beyond the region of the proboscis, but was unable to find the trumpet-shaped tubes. These are probably situated in the posterior segments, as Professor Verrill has seen the male worms discharging their milt from that portion of the body. The fecundation takes 2:)lace in the water. EXPLANATION OF PLATES. Plate XLII. Figure 1. — Nereis virens, female; dorsal view of the anterior portion of the body; a, head, with four eyes ; 6, h, antennte ; c, c, palpi ; c', c', lobes of palpi ; cZ, buccal ring ; ee, ee, longer dorsal pair of tentacular cirri ; ee', ee', shorter dorsal pair of tentacular cirri ; e, e, longer ventral pair of tentacular cirri ; e', e', shorter ventral pair of tentacular cirri; h, lateral appendages; g, abdominal rings. Fig. 2 and 2a. — Two forms of setaj; a, shaft; h, blade. Pig_ 3. — First lateral appendage of female, posterior view ; lettering the same as in fig. 8. Fig. 4. — Second lateral appendage of female, posterior view ; lettering the same as in fig. 8. -pig, 5. — Third lateral appendage of female, posterior view ; lettering the same as in fig. 8. Pig_ 6. — Fourth lateral appendage of female, posterior view ; lettering the same as in fig. 8. Pig. 7. — Fifth lateral appendage of female, posterior view ; lettering the same as in F. M. TnrnbuU — Anatomi/ and Habits of Nereis virens. 279 Fig- 8. — Fortj'^-fifth lateral appendage of female, posterior view; A. upper ramus; B, lower ramus; a, dorsal cirrus; &, superior ligula of upper ramus; c, anterior setigerous lobe : /c, posterior setigerous lobe ; d, inferior ligula of upper ramus ; e, e,' divisions of anterior setigerous lobe of lower ramus ; /; posterior setigerous lobe; f/, inferior ligula; h, ventral cirrus; i, shoulder of upper ramus; s, s', sets ; y, y', aciculEe. Fig. 9. — One hundred and nintli lateral appendage of female, posterior view ; lettering the same as in fig. 8. Fig. 10. — First lateral appendage of male, posterior view; lettering the same as in fig. 8. Fig. 11.— Fifth lateral appendage of male, posterior view; lettering the same as in fig. 8. Fig. 12. — Forty-fifth lateral appendage of male, posterior view; '•■, extra division of anterior setigerous lobe ; x, shoulder, peculiar to the male, on the dorsal edge of the lower ramus ; otherwise the lettering is the same as in fig. 8. Fig. 13. — One hundred and ninth lateral appendage of male, posterior view; lettering the same as in fig 8. Plate XLIII. Fig. 15. — Nereis virens; ventral view of the head and mouth, the proboscis withdrawn; '7, head ; b, h, antennae ; c, c, palpi ; c', c', lobes of palpi ; d, buccal ring ; ee', ee', shorter dorsal pair of tentacular cirri ; e, e, longer ventral pair of tentacular cirri ; e,' e,' shorter ventral pair of tentacular cirri ; ?n, mouth ; g, abdominal rings ; h, h, lateral appendages. Fig. 1 fi. — Nereis virens; the walls of the body are cut through longitudinally on the dorsal side, so as to show the perivisceral cavity with the alimentary canal; m, mouth; «, muscles of mouth ; x, pharyngeal region of the proboscis ; c, muscular partitions of proboscis; 6, perivisceral cavity: p, dental region of proboscis; o, oesopha- geal region of proboscis ; j. salivary glands ; r, intestine proper ; a, muscular partitions. Fig. 17.— Head of Nereis virens, with the proboscis protruded, dorsal view; a, head; b, b, antennae ; c, c, palpi ; c', c', lobes of palpi ; d, buccal ring ; ee, longer dorsal pair of tentacular cirri ; ee', ee', shorter dorsal pair of tentacular cirri ; e, e, longer ventral pair of tentacular cirri ; e', e', shorter ventral pair of tentacular cirri ; /, /, jaws; 0, anterior median area of dorsal side; I, I', left and right anterior sub- median areas of dorsal side ; w, «', anterior lateral areas ; i, posterior median area of dorsal side ; s, s', left and right posterior sub-median areas of dorsal side ; ?•, posterior lateral and ventral areas. Fig. 18. — Protruded proboscis, ventral side; /, /, jaws; w, w', anterior lateral areas; y. anterior median area of ventral side ; x, %', left and right anterior sub-median areas of ventral side ; r, posterior lateral and ventral areas. Fig. 19. — Jaw of Nereis virens, much enlarged. Figs. 27 and 28. — Nervous system o( Nereis virens ; h, abdominal ganglia; 7i, n, nerves to muscular partitions ; o, o, nerves passing through partition to preceding segment ; TO, m, nervous trunlcs to feet ; k, k, ganglia sending off a cutaneous branch and a branch (i) supplying nerves to tlie feet ; d, d, connectives ; d', d', accessory connect- ives ; g, g, nerves communicating with the ganglia of the proboscis ; e, e, ganglia sending branches to the internal or ventral tentacular cirri ; e', e', ganglia sending 280 F. M. Turiibull — Anatomy and Habits of Nereis vlrens. brancbes to the external or dorsal tentacular cirri ; a, head-ganglion with four eyes ; h, b, nerves to antennae ; c, c, nerves to palpi ; iv, iv, series of ganglia and nerves on the ventral side of proboscis ; /, /, jaws, ventral side ; ?/, y, the series of ganglia and nerves on the dorsal side of the proboscis; / / jaws, dorsal side. In figure 27 the proboscis has been revolved about a line passing through g, g, so that the ventral side is uppermost. The head and abdominal ganglia are in their nat- ural position. The position of figure 28 is reversed so as to show the relations of the dorsal ganglia y, y, to the ventral ganglia w, w (fig. 27). Plate XLIV. Fig. 20. — Circidation of blood in Nereis virens, and also the alimentary canal in its natural position ; to, mouth ; x, pharyngeal region of proboscis ; p, dental region of proboscis; o, oesophageal region of proboscis; r, intestine, covered with a vascular network, which is connected in each segment with the large dorsal and ventral vessels by short Ijranches ; j, salivary glands ; a, large dorsal vessel ; b. large ventral vessel; c, c', c'', peripheral vessels; d, branches from the dor- sal side of the feet; e, branch to oesophagus ; /, /'', branches to the ventral side of the feet; s, lateral branch, supplying the vascular network {g) and first three lateral appendages; A", vessel from network (g) to base of tentacular cirri; t, branch from the vascular network (u) on the proboscis, to the large ventral vessel (6) ; V, branch from the large dorsal vessel (a), at base of head, to the network (u) on the proboscis. Fig. 21. — Diagram, showing the disposition of the large ventral vessel, and its branches on both sides, in the anterior portion of the body ; I, continuation of the large ventral vessel along the median ventral line beneath the muscles of the proboscis ; n, vascular ring surrounding tlie pliaryngeal region of proboscis ; otherwise the lettering is the same as in fig. 20. The arrows indicate the direction in which the blood flows. Pig. 22. — Diagram to show the circulation of the blood, and also the relative position of the parts, in one segment of the body ; i, intestine ; p, perivisceral cavit_y ; h, crypt from which aciculee grow ; 1. muscles of crypt, which are attached to the base of the foot ; g, g, segmental organs ; k, walls of body ; n, a ganglion of the abdominal chain; «, large dorsal vessel; b, large ventral vessel; c, peripheral vessel ; /, branch to ventral side of foot ; d, branch from dorsal side of foot. Fig. 23. — Circulation of the blood in the last posterior segment ; a, large dorsal ves- sel ; 6, large ventral vessel ; c, vascular ring, with no branches. Fig. 24. — Lateral view of the circulation in one segment; d, branch from the dorsal side of foot: /, branch to the ventral side of foot in the adjacent posterior segment. In the last two figures arrows indicate the direction in which the blood flows. Fig. 25. — Dorsal view of two segments showing the vascular network in the lateral appendages and beneath the skin of the back; lettering the same as in fig. 22 ; the large dorsal vessel, a, and the peripheral vessels c, c, show through the trans- lucent walls of the body. Fig. 26. — Segment showing the vascular network beneath the skin of the ventral side and in the lateral appendages; lettering as in fig. 22. VII. Median and Paired Fins, a Contribution to the His- tory OF Vertebrate Limbs. By James K. Thacher. Median Fins in Amphioxus. The quadrate markings seen at the base of the median fin in Amphioxus extend on the dorsal side from one extremity of the animal to the other, or nearly so, and on the ventral side from the porus abdominalis aboi-ad toward the extremity of the tail. They are largest and most distinct in the middle of the body, and become smaller and less clearly marked (as seen from the outside), toward the head and tail until they seem to fade out entirely as they closely approach these extremities. As Stieda* has shown these are but the external marks of a series of cavities, containing what is described as " a transparent, wholly struc- tureless mass, resembling a coagulation." Thus the relation of these bodies to the somewhat similarly placed primordial tin rays, or "interneural spines," of the Craniote fishes is not so simple and direct as indicated in the mistaken repi-esentations of Rathkef and of Mtiller.J Still they occupy a position similar to that of the primordial fin rays of other fishes, and the fact that they di» not agree with the segmentation of the lateral muscles, seems to have some pertinency here, and to this alone I wish to call attention. In the middle of the back there are about five of these bodies to a single segment, and on the ventral side just aborad of the abdominal pore there are about four to each. As we shall see hereafter, the structures of the median line (genuine fin-rays except in Amphioxus) exhibit quite generally throughout the fishes, a total disregard of the segmentation of the lateral muscles, and are more numerous than those segments. * Studien iiber den Amphioxus lanceolatus von Dr. Ludwig Stieda, Mem. de I'Acad. Imp. des Sciences de St. Petersbourg, VII^ Serie, Tome xix, No. 7. f Rathke, Bemerkungen liber den Ban des Amphioxus lanceolatus. Konigsberg, 1841. X Johannes Miiller, Ueber den Bau and die Lebensersclieinungen des Branch iosioma lubricum. Abhandl. der Berliner Academic, 1842. Trans. Conn. Acad., Vol. III. 36 February, 1877. 282 ./. K. Tharher — MedUm and Paired Fins. Median Fins in Myxina. In Myxine 'glutinosa the median tin extends but a short distance forward. In a specimen 24 centimeters long, from the Bay of Fundy, the fin reaches 4 cm. from the extremity of the tail on the dorsal side, and 2*5 on the ventral. The fin-rays, now unquestionable homolognes of the ])rimordial fin- rays of Gnathostomes, tliough not yet having assumed the histologi- cal structure of true cartilage, support the thin fold of skin which forms the fin. They are simple tapering rods, extending distally to the edge of the fin, and proximally scarcely dipping below the general body contours. The only deviation from sim])le rods which I have been able to find is the dichotomous splitting of some of the rods where the fin rounds the extremity of the tail. The numerical relation between these rays and the corres})onding muscular segments is as three to one on the dorsal side, and as two and a half to one on the ventral. I have been iinable to detect any muscular fibers in the composi- tion of the fin. Median Fins in Petromyzon. Here the median fins are much better developed. In a specimen (^Petromyzon marinus, from the Connecticut River), 77 cm. long, the caudal fin extends forward along tlie dorsal side 7*5 cm., sloping downward nearly to the body, then tlie second dorsal rises abruptly and runs orad 16 cm., where it reaches by a gentle slope the general out- line of the body. There follows a finless space ;1"5 cm. in length which is succeeded by the first dorsal, whose extent is 9 cm., being therefore shorter as it is lower than the second dorsal. The anus is opposite the orad part of the second dorsal. The fins, therefore, take up almost the whole of the hinder half of the mid-dorsal line. In Myxine only one-sixth was thus occupied. On the ventral side we have only the caudal, extending about as far here as it does above. These fins are sup})orted by a series of chondroid rays, lying quite close to one another in the median plane. They are straight and slope aborad from the fatty-fibrous ridge-pole of the myelonal canal (PI. XLIX, fig. 1, a,) to the very edge of the fin. They are found in all the fins. Their form is represented in PI. XLIX, fig. 1, where only one ray is drawn complete. As shown, it bifurcates twice and thus ends distally in four fine branches. This figure is IVom the central p.'irt of J. K. Thdcher — Mediati and Paired Fins. 283 tlu' lartio second dors.al, and sliows one of the longest rays. Where they are shorter we may liave only one bifurcation, or one of the two l)riniary branches, that toward the longer rays, may again divide, while the other toward the shorter remains uncleft. Farther forward at the beginning of the second dorsal, where the rays are still shorter, they do not divide at all, but end somewhat bluntly though com- pressed from side to side. Each ray is largest in the middle and here lies quite close to its adjacent rays ; below they grow more slender, and therefore are some- what spaced, but expand somewhat to a foot resting on the myelonal canal. I have seen no cases of concrescence between adjacent rays. With the exception of the variation in the branching and length in different parts of the fin, before alluded to, the rays are all similar and parallel one to another. This branching is plainly a true dichotomy and not the product of concrescence, as is evidenced by the total absence of anything else resembling concrescence, by the similarity in size between two adja- cent differently branched rays, and by the regularity of the branching. On each side of the row of skeletal elements are muscular bundles of a somewhat blacker color than the two great masses of lateral muscles. The muscles of the median fin wedge themselves into the angle between the lateral muscles of the two sides along the median line. They ai-e sharjdy distinguished from these. The fibers of the lateral muscles run longitudinally, while those of the fin mus- cles are parallel to the pi'iniordial median fin-rays. There is abso- lutely no continuity between the two in any part. Moreover the bundles of the fin-muscles show no relations to the segments of the lateral muscles. A cross section, PI. XLIX, fig. 3, shows the relation between the fin-muscle bundles and fin-rays. The numerical relation between the fin-rays and the segments of the lateral muscle is shown in PI. XLIX, fig. 2, where we have a little less than four of the former to one of the latter. The relation between the tin-rays and the neural arches is shown in figure 1, where we have 35 rods and 23 arches. These neural arches rise from the sheath of the notochord, to stiffen the fibrous sides of the myelonal canal and to apply themselves to its fatty- fibrous ridge-pole. The fin-iays abut on the same ridge-pole in the mid-dorsal line, but they are in no way connected with the neural arches. 1 have met with no cases even of concrescence between the two. 284 ./ K. Thacher — Median and Faired Fins. Figure 4, PI. XLIX, shows the relation between the neural arches and the segments of the lateral muscles. There are two arches to one seo-ment. In fio-ure 1, then, there must have been lU segments, which gives us a very little more than three as the ratio between the fin-rays and the segments. From figure 2 we obtained a little under four. Both results are necessarily correct. There is considerable variation in the relation between the fin-rays and the muscular seg- ments and the neural arches. This is exhibited in the following table of observations on a single specimen. Fin-raj's. 19 Arches. Katio of fln-rays to arches. Ratio of fln-rays to segments. In the orad half of ID. 8 2-4 4-8 In the aborad half of ID. 20 8 2-5 5- In the orad part of 2D. 20 11 1-8 3-6 In the next 35 rays of 2D. 35 23 1-5 3- In the next 1*1 rays of 2D. 27 17 1-6 3-2 In the next 12 rays of 2D. 12 7 1-7 3-4 The Lampreys then have advanced beyond the Myxines toward the Gnathostomes as regards the structure of the median fins, in the greater development and efficiency of those organs, and in the addi- tion of special fin-muscles whioh seem to be wanting in the lower group. And while in Myxine the original independence of the median fin skeleton and the axial skeleton is shown by the e.vistence of the median fin-rays before any neural arches have appeared, and by their want of agreement with the musi-ular segments to which the axial skeleton will conform when it does arise, in Petromyzon we have the same independence more strikingly reaflirmed by the simul- taneous existence of neural arches and median fin-rays, and their perfect indifference to one another. Median Fins of Elasinobranchii.* We shall here consider the skeleton of the dorsal and anal fins alone. That of the caudal fin has on the ventral side undergone peculiar modifications by the union of tin-rays with hnemal s[)ines. * The term Elasmobranchii includes the Chimaeroids with the Sharks and Rays. I have had no opportunity of examining the tin skeletons of the former. But Chimaera seems to me to be but a divergent form of Sharks and to have its nearest living relative in Cestracion. They are quite specialized forms, Cestracion the less so. I will state here that the Elasmobranchs described came from Wood's Hole, Mass. The names used are those given by Dr. Gill in the IT. S. Fish Coram. Rep. for 1871-72. J. K. Tharhr)' — Median and Paired Fins. '285 In general the ^<l^u•tul•e of tlie medi.-ui tin resembles wliat we have seen in Petromyzon, but there are important differences. The rays are of hyaline cartilage and they do not usually reach down to the ridge-pole of the myelonal canal. In EKlamla, for example, in tlie smaller of the specimens %ured, the rays approach within a centime- ter and a half at the oiad extremity of the first dorsal fin, but are three and a half centimeters distant at the other extremity. In Si/ualus {Ae(()ifh)as) they come closer. This ridge-pole consists of a cord of rather peculiar white lono-i- tudinal fibers, constituting now a '' liyauieatuin longltitdinale.'''' It appears to me undoubtedly homologous with the fatty-fibrous body in Petromyzon. The cartilaginous arches unite under, and do not extend around over it, though they clasp it somewhat. The rays are segmented, usually twice. Dichotomy is rare if not altogether absent. Concrescence of adjacent rays is by no means uncommon. The reduction of rays in size is exhibited in all deo-rees. Calcification presents itself in a thin superficial layer on each side of the somewhat flattened ray, but it fails on the edges, i. e. as we come close to the median plane. The muscles of the fin, as in Petromyzon, are in total independ- ence of the large masses of segmented lateral muscles, but they are in more definite relation with the skeleton of the fin. This is accu- rately represented in PI. LIX, fig. 66, though that is a s-;ction of a pectoral and not of a median fin. We see that each ray has on each side a special muscle, separated from its fellows by the fibrous sheet which runs from between the rays to the integument. Each little muscle developes in its median line a flat tendon, which, parallel to the surface of the fin, inserts itself in the fascia covering the extremi- ties of the fin-rays and the proximal ends of the well known horny fibers, which here supplement the primordial skeleton, as the second- ary fin-rays of Ganoids and Teleosts do. The relation between the niimber of fin-rays and that of the vertebra? opposite to them is similar to what we ha^e seen in the lower forms. In the Nictitantes, for example, there are on the average about '2-5 rays to one vertebra. But there is considerable variation, even in individuals of the same species. The extreme numbers, so far as I have observed, are 3 '5 in an anal of Sphyrna, and 2 in a first dorsal of Eulamia. We turn now to the more minute examination of several species.. 286 ./. K. ThacJier — Median and Paired Fins. First Dorsal of Mustelus cants, PI. XLIX and L, figs. 5-10. Id pi. XLIX, fig. 5, we have 24 separate rods, unless ?> be the terminal piece of 2 ; but its conformation seems to testif}' to its independent but reduced character. The number here then is 24 or 2,3. In PI. XLIX, fig. 6. we have again the same alternative, witliout quite so strong a case for 24, but still quite strong. In PI. L, fig. 7, we have the choice between 24 and 2.5, but in favor of the latter. In PI. L, fig. 8, we have 23 or 24, but the former has the greater probability. PI. L, fig. 9 gives us 22, 23 or 24, 23 being more probable. PL L, fig. 10 exhibits 23 or 24, dependent on the view taken of rays 5, 6, and 7. The probability seems in favor of 24. I think we may sum up then with regard to the number of rays constituting the first dorsal of Mitslelus canis thus : it has usually 24 rays but this may vary to 23 or 25. Nearly all the rods are segmented twice. The distal line of segmentations fails in the one or two orad; and the proximal, in the two or three aborad ones. Additional segmentations are ver}' rare. What might be reckoned as such are seen in fig. 7, ray 4 ; fig. 6, rays 5 and 6. This makes an average of ■() + .* The union of adjacent rays is rare. I estimate it at '04 of the total possible con- crescence.f Shortening or reduction in size is likewise rare. We have first those questionable cases of which fig. 5, rod 3 seems the least questionable ; and then plainer but less extreme instances in fig. 7, ray 5; fig. 9, ray 18; fig. 10. ray 7, then we have the usual shortening of the rays at the extremities of the fin. When these aborad rays shorten up, those next in front of them have a remarkable tendency to grow up under them. This is well shown in fig. 9. iforeover when in the aborad rays the proximal joint becomes very short, it is some- times divided into two lateral halves. This is the case for example in rod 22 of fig. 10. Occasionally we have a minute piece or pieces of cartilage forming a tip to a ray. It cannot act as an extra joint, by giving increased flexibility to the ray. And it seems doubtful whether the origin of the two is to be referred to the same causes. Yet intermediate forms occur so as to raise the question whether they are to be referred to one or the other category. Tliese tips seem to ))e exhibited in fig. 7, ray 4 ; fig. 8, rays 2 and 3 ; fig. 9, ray 3. This gives „ — - — 0 + . I now find the ratio of the proximal piece of the middle ray of each tin to the middle piece of the same. The average of these ratios is -6. The method gives -3 as the ratio between the terminal and middle joints. We have then for the first dorsal of Mustelus canis : Number of rays 24. Extra segmentations '0 + . Concrescence '04. Betipping -0 + . Katio of proximal to middle piece of middle ray -6. Ratio of distal to middle piece -3. * The decimal is obtained by dividing the number of additional segments by the number of rays. \ The amount of concrescence between two adjacent rays is the ratio between the length of the union and the whole distance through which they are adjacent and might have united. The sum of these fractions divided by the number of rays less the number of fins, gives the estimate of the concrescence. J. K. TharJier — Median and Paired Fins. 287 Second Dorsal of Mvstelus cams, PI. L and LI, figs, ll-lf). Fig. 11 has 24 rays. fig. 12 has 22. fig. 13 has 2:i, fig. 14 has 24, fig. 15 has 24. Tluis of the five examined three have 24, one 23 and one 22 rays. It seems probable that a wider examination would give as forms having 25 or more rarely 2(). Thus we have for the second dorsal 24 rays with some slight variation. Extra segmentations appear only in fig. 11, ray 5 ; and fig. 14, ray 2. Tliis gives -0 + . I estimate the concrescence at -06. The concrescence is mostly confined to tlie proximal row, and is more frequent at the two ends than in the middle of the series. Betipping is seen only in fig. 14, ray 23. This gives ^0 + . Ratio of proximal piece of middle ray to middle piece '6. Ratio of distal piece of middle ray to middle piece -4. Tlie downward prolongation of the proximal parts of one or two of the orad ra^-s is noticeable, being quite pronounced in all the cases except that represented in fig. 1 1 . The reduction of the rays is rare, but shown to an extreme extent in fig. 11, ray 1 ; and fig. 15, ray 1. Anal of Mustelus canis, PI. LI and LII, figs. 16-19. Figs. 16 and 17 have each 18 rays. Fig. 18 has IT or 18 according as the last ray is double or not. Fig. 19 has 18 or 19 under the same conditions. The great width of the last ray in the last two cases makes the larger number probable. Thus we have 18 as the normal number, with probably slight variations. Extra segmentations are seen in fig. 17, ray 5 ; and fig. 19, ray 3. This gives '0 + . The concrescence I estimate at -09. Betipping occurs in fig. 16, ray 3. This gives "0 + . Ratio of proximal piece of middle ray to middle piece is -I. Ratio of distal piece of middle ray to middle piece is "6. First Dorsal of Galeocerdo tigrinus, PI. LII, fig. 20. In this sole specimen there are 25 rays. Extra segmentation occurs in 10, 20, 21. 22, 23, which gives -2. It should be noticed that this extra segmentation is in each case here a doubling of the proximal line of segmentations. Concrescence is estimated at -06. Betipping is seen in 18, 19 and 25. This gives '1. Ratio of proximal piece of middle ray to middle piece 1-1. Ratio of distal piece of middle ray to middle piece -6. Shortening is seen in 6 and 22. In the latter the proximal piece is ext luded from the edge of the fin by a, the proximal piece of 2;!, and by the proximal piece of 21. The piece a consists of two lateral halves. Second Dorsal of Galeocerdo tigrinus, PL LII, fig. 21. Number of rays 13. Extra .segmentation in 3, giving -1. Concrescence is estimated at -01. Betipping, in 5, 8 and 13, gives -2. Ratio of proximal piece of middle ray to middle piece 1-3. Ratio of distal piece of middle ray to middle piece -6. Anal of Galeocerdo tigrinVjS, PI. LII, fig. 22. Number of rays 12. Extra segmentation in 6, 7. 8 and 10. gives 3. Concrescence is estimated at -05. Betipping in 4 gives "l. Ratio of proximal piece of middle ray to middle piece I'l. Ratio of distal piece of middle ray to middle piece -7. 288 ./ K. Thacher — Median a7id Paired Fins. First Dorsal of Eulamia Milherti, PI. LII and LIU, figs. 28 and 24. Number of rays 28 or 29. ^ As indicated by the numbering of the rays, I take number 21 in each figure to be a single ray, which has widened at the top, and been segmented in the way figured. Extra segmentations in fig. 23. rays 20 and 21 (2 extra segmentations in the latter) in fig. 24, rays 4 and 21. This gives •!. Concrescence is estimated at -09. Betipping is absent. Ratio of proximal to middle piece •9. Ratio of distal to middle piece '9. Second Dorsal of Ealamia Milherti, PI. LIU. figs. 25 and 26. The number of rays differs remarkably in the two specimens, being 12 in the one and 16 in the other. It must, however, be remembered that the second dorsal has become very small and of very little physiological importance. Organs which have thus become functionless are peculiarly prone to vary. They thus secure more easily some other and new function. We will take the average number 14 as the normal one for the rays of this fin. Extra segmentation occurs in fig. 25, ray 6 (twice), and in fig. 26, ray 3 This gives •!. Concrescence is estimated at -09. Betipping is absent. Ratio of proximal to middle piece of middle ray -8. Ratio of distal to middle piece of middle ray -5. Anal of Etdamia Milherti, PL LIII, figs. 27 and 28. Number of rays 17 or 18. Extra segmentations in fig. 27, rays 7, 10 and 12 (twice in the latter); in fig. 28, twice in 12, once in 16, give 2. This implies a certain interpretation of the ambiguous rays 11 and 12 in figure 27. In fig. 28 we seem to have a plain case. Here the ray 1 2 is broadened at the top, and its distal piece divided by two intersecting cuts into four pieces. Ray 12 in fig. 27 is explained in the same way. Ray 11 is a little shortened, and excluded from the edge by the tips of 10 and of 12. This appears to me the most probable view of the case. Concrescence is estimated at ■12. Betipping absent. Ratio of proximal to middle piece of middle ray -7. Ratio of distal to middle piece of middle ray -4. First Dorsal of Sphyrna zygcena, PI. LIII and LIV. figs. 29 and 30. Number of rays 33 and 34. I regard the three pieces at the extremity of 28 as belonging to that ray. It has been widened and divided like the instances in Eula.mia. Extra segmentations, fig. 29, rays 2 and 3, twice; rays 4 and 5; ray 28, twice; fig. 30, ray 28, twice, give -2. Concrescence is estimated at -07. Betipping absent. Ratio of proximal piece of middle ray to middle piece 7. Ratio of distal to middle piece of middle ray 2-4. In fig. 29 the proximal line of segmentation fails in rays 9-17, except in the joined rays U and 12 where it is present. In fig. 30 it fails in rays 8-16. J. K. Thacher — Median and Paired Fins. 289 Second Dorsal of Sphyrna zygcena, PL LIV, tig. 31. Number of rays 14. Extra segmentations amount to 10. Concrescence is estimated at '07. Betipping is absent. Ratio* of proximal to middle piece of middle ray -3. Ratio of distal to middle piece of middle ray -3. The last ray, both in the second dorsal and the anal, is large and round. Anal of Sphynia zygana, PI. LIV, fig. 32. Number of rays, 27. Extra segmentations 8 (1), 9 (1), 10 (1), 11 (1), 12 (1), 13 (1), 14 (1), 15 (1), 16 (1), 17 (i), 18 (2), 19 (1), 20 (1). This gives "5. Concrescence is estimated at -03. Betipping none. Ratio of proximal to middle piece of middle ray -4. Ratio of distal to middle piece of middle ray -3. First Dorsal of Eugoynpliodus litoralis, PI. LIV, and LV, tigs. 33-39. Specimens figured in tigs. 33, 34 and 36 have plainly 16 rays. Those in figs. 37 and 39 have plainly 17. Those in tigs. 35 and 38 have 16 separate rays, but the last is quite broad. Where we have plainly 17 rays, figs. 37 and 39, the last two rays have united with the exception of their distal joints. We may fairly conclude that figs. 35 and 36 present a more complete concrescence of those rays. We have then as the number of rays 16 or 17, the former in three cases, the latter in four. Extra segmentation is estimated at '7. Concrescence is estimated to be "05. Betipping reaches -4, each separate piece being counted. These small nodules of cartilage sometimes seem very evidently to be a continuation of a ray upon the fol- lowing ray. But not infrequently they seem to be scattered rather irregularly along the edge of the fin. It will be noticed that they are most frequent in the orad part of the fin, though not on the first two or tliree rays. It is very probable that the estimate of their frequency should be higher than given, for they are easily lost in the preparation of the specimen. Ratio of proximal to middle piece of middle ray -5. Ratio of distal to middle piece of middle ray -4. Second Dorsal of Eugomphodus lltoralis, PI. LV and LVI, figs. 40-46. In fig 42 we have 16; in 45, 17 ; in 40 and 46 we have 18 rays. These are all plain cases. Fig. 43 exhibits 17 rays, but raises a suspicion of 18 by the breadth of the last ray. Pig. 41 gives 16 or 17, probably the latter. Fig. 44 leaves us in doubt between 14, 15 and 16, with, as it seems to me 15, the most probable. We may take 17 as the normal number. As far as the evidence here goes the second dorsal is more liable to vary than the first. We see that in each the greater the number of rays, the greater is the amount of concrescence. Extra segmentation amounts to "6. Concrescence amounts to -10. Betipping amounts to -4. Ratio of proximal to middle piece of middle ray is '3. Ratio of distal to middle piece of middle ray is -3. * Where a segmentation is double the point half way between the joints is taken as the limit between the middle and extreme piece. Where it is triple the middle seg- mentation is taken. Trans. Coxn. Acad., Vol. III. 37 February, 1877. 290 ./. K. T/iiichn' — Median a ml I'dircd hlnx. Amil of Eugompliodua Ulorulis, 1*1. L\'l and LVIl, (i<;'s. 17-5(1. The number of rays is 21 in figs. 47 iiiul -49. In 18 \\v liavo 'I'l, and in 50, 'JO rays. We may take then 21 as the normal nunilicr. Extra sogmoiitations amount to 1. ConcTOSoonco is estimated at ■](). Hetippin^- amounts to •:{. Ratio of proximal to middle piooo of middle ray i. Ratio of distal to middle pieee of middle ray 5. Wo may sum up the results of this investigation of the reseiul)lani'es and dillercnces of the forms so far examined in the following table. No. of Kxtra ( ion- rays. seKliU'iitatioiiH. (■iH'HC'ciicc. Brli|i|iiii; Katio of Hatlo ot |.ro\. to (Uwtal to iiild. mid. i>ic('(^ pk'co of mill. )riiii(l.rHV. rav. First Dorsal. Second Dorsal. Anal. i'lugomphodus . . 17 Mnstelus 21 (lalooeerdo 25 K\damia 29 Spliyrna :M i^yugomphodua _ . 17 Mnstelus 21 (lalooeerdo 1 :i Eulaiuia II Sphyrna -II ' Mugomphodus ..21 Muslelus 18 Oaleoeordo 12 Mnlamia 18 Spliyrna 27 •7 •05 •1 •5 •1 •(I •01 ■0 •() •;{ .■) •in; •1 11 •(5 •1 ■oil ■0 •!» •9 •(I ■07 •10 ■II ■1 •7 •;; •0 ■1 •oc ■01 ■0 •(; 1-:! •4 (•> •1 •01) •0 •8 ■5 •(I ■07 ■0 •:{ •.•{ •1 ■10 •;; •4 •5 •u ■0!) •05 •0 •1 •7 11 ■(; •7 •2 •12 •0 •7 ■4 •5 ■o;i •0 •1 ■;{ Tlie eh:ini]fos whicli ;ire presenUHl consist chieHy in concrescence. This takes place in various ways. The proximal portions more fre- quently vmite tlian the distal, but we may have conci-escence of the distal points while the proximal portions are separate. The reduction of rays is exhibited in all deoreos from the slightest shortenino- t() the extreme degree shown in PI. LVTI, tig. 50, ray 0. It is perhaps noteworthy that the changes which would have suf- ficed to differentiate the fin of one species from another, if they had continuously advanced for a few generations, are changes which in no slight degree are now taking place ln'twcen parents and childi-cn. I mean changes of number, segmentation, concrcsccni'c, and relative lengths of parts of rays. I exhibit SOUK' figures of the dorsal tins of Sqin/li/.^, 1*1. LVIT, figs. 51-r)0, and PI. LVIII, iig. 57, liditt, PI. LVIII, tigs. 5S, 55), and Mt/liohafiti, PI. 7>VIIT, fig. (iO. It is perfectly certain that they have been derived from a sei-ies of parallel rays, and that the jjrincipal process of change has beiMi in tlu' way of concrt'scence. They tell their own story as far as it can be told without the investigation of other closely related forms. The last ray in the first dorsal of /xa/'o, PI. lAIII, tig. 5S, is remarkable, and it is represented in the second doi-sal. fig. 50, by a serii'S of se])arale nodules. J. K. TJiacher — Medhin (iiul Paired Fins. 291 I m1s(. add tiouivs of the dorsal, I'l. LVIIF, fig. 01, and anal, PI. LIX, lio-. 02, of A(-/j)enNer. Tlu'so arn very similar to the simpler shark dorsals. It will be iioticed tliat we have the ])redoniinant division into three pieces, hul llie terminal piece is very shoi't. (Joiiiiiisioiis rei/arditKj McxVihh Fins. 'i'lie priniortlial median lin-rays in whatever form they oce\ir are derivatives from a serit's of simple parallel chondroid rods, which grew up in the me<lian fold in total independencte of the cartilaginous arches alxni' and Itelow the notochord. These earliest representa- tives of these parts weri' irom two to four times as numerous as the vertebra' opposite them. In the (inathostomi true hyaline cartilage replaced (he lowei- form of tissue seen in Myxlne and Petromyzon. Segnu'ntation and c(»ncresceiice, as well as redu(!tion in size, were common changes in the (rnathostomes, and here a division into three parts is the usual though not invariable rule. Hence it is seen that (iegenbaui"'s* st:itement that, in their sinijjlest forni,^the prinioidial tin rays are mere pi-olongations of the neural spines is incon-ect. It has been demonstrated that this was not the earliest foi-m. The l)ij)noans, however, seem to oft'er an example where the priim)rdial meclian fin-rays are mere prolongations of the neui-al spines. They demand a moment's consideration. True neural spines are lirst found in the (ianoids. They are absent in the Klasmobranchs and Agnathostomi. PI. LIX, fig. 0;i represents the projection of a section of a vertebi-al segment of Acipenser cut through the middle line of the arch ami neural spine. .\s the latter slopes backwai'd, it is considerably fon'shortened in the figure. Now the cartilaginous arches spriiigiiig from the sheath of the notochord pass upwards to lay themselves on each side of the fibrous cord />, and here they sprea<l inward to meet oiu' another on the ventral side of />, and also pi'olong themselves above to almost or quite meet, and then they are followed by the dorsal spine a, from which they are se])arated by a segmentation. Now the cord A, which is the same as the Uyainentum loiKjitadi- * Griiiidriss der Vergleichenden Anatomie, 1873, p. 488. Gregenbaur's assertion in the .same place tliat they usually correspond in number to the vertebra? opposite, is very strange. We have already seen that they do not do this in the earlier and more significant forms ; and the statement of Gegenbaur would decidedly misrepresent what we find, for e-xamplc, hi tlie figures of fish skeletons in Agassiz's PoisBons Fossiles. 292 J, K. Thacher — Median and Paired J^)'ns. nale in Cerafodus* is also, without the slightest doubt, homologous with the chord which lies entirely above the neural arch in the Elas- mobranchs. The peculiar fibrous character is almost exactly the same in each. We have seen that this cord in the sharks is in all probability homologous with the fibrous tatty ridge-pole of the neural canal in Petroniyzon. Thus while in the latter the arches of one side and the other are entirely separate, in the sharks they have spread beneath the ligament so as to meet, and in the higher Gnathostomes they have also joined above it, or nearly joined, for the origin of a is still to be discussed. There are two possibilities with regard to the neui-al spine a. Either it is formed by the union of a median fin-ray with the neural arches, the ray thus constituting the keystone of the arch, or else by the union of the neural rods from each side and their prolongation dorsad. But the junction between <i and h is quite close ; the neural spines correspond in number and j)osition with the lateral parts of the arch ; while fig. 61 shows conclusively the absolute independence of neural spines and primordial median fin-rays. The second of the two possibilities is then the true one. Thus neither are median fin-rays derived from neural spines, nor neural spines, where they occur, from primordial fin-rays. But the cartilaginous supports of the median told in the Dipnoans are very long and segmented. They are simply elongated neural spines and are not primordial fin-rays in any homological sense. If they were formed by the reduction in number of the primordial fin-rays and their coalescence with the neural sjjines it is impossible that we should not have here and there an extra one, and some evi- dence in the case of others of such a junction. But there is nothing of the kind, either in the descriptions of Gtinther in the case of Gera- toduSjj or in those of Owen J and Peters§ in that of Protopterus annec- tens. or in those of Bischofi"|| in that of Lepidosiren paradoxa. Gtln- * Griinther's Description of Ceraiodus, Pliil. Trans., vol. elxi, pt. ii, PI. XXXVIII, Pigs. 3-9, e. f Phil. Trans., vol. clxi, pt. ii, 1871. G-iinther, Description of Oeratodu.s. In Giinther's fig. 2, PI. XXX, the proximal joint of the 14tli neural spine seems to bear two ' ' interneurals," one orad of the other. But as no notice is taken in the text of this, which would be a very noteworthy fact, if it were fact, and as the description of these parts there given is such as would demand a notice of this exception, it is evident that it must be an inaccuracy in the figure. :|: Trans. Linnean Soc, vol. xviii, pt. iii. Owen, Description of Lepidosiren annedens. § Miiller's Archiv., 1845. Peters, Ueber einen dem Lepidosiren annectens verwandten, Fiach von Quellimane. II Ann. Sc. Nat., xiv, 1840. Bischoif, Sur le Lepidosiren iiaradoxa. J. K. Thiieher — Median and Paired Fins. 298 tiler's denoraiiiation, tlu'ii, of tlie ultiniato and ])eiiultimate joints of the neural S])ines of Ceratodas as " interneural first" and " interneural second," is ill chosen, and rests on a mistake in homology. That great genetic group, then, consisting of Dipnoi, Amphibia and Am- niota, seems to have entirely lost those primordial median fin-rays which appeared so early and are found even in Mxjxine. Limb-skeleton of Air breathing Vertebrata. In 1864, Gegenhaur* set forth the splendid results of a widely extended investigation of the limb-skeleton of the air-breathing vertebrates. Herein was established the typical form of these parts for this large gi'oup, consisting of Amphibia and Amniota. Inasmuch as there is no doubt of the natural, that is the genetic, character of this group, and inasmuch as it is marked out from all other vejte- biates by the development of a fenestra ovalis and the modification of the proximal part of the second post-oral, or hyoid, arch into a stapes in connection therewith, I ventui'e to use the name Stapedifera in place of the circumlocutory air-breathing Vertebrates. For the Stapedifera, then, the typical limb-skeleton was established ; typical in the sense of the older anatomists, as that ideal form from Avhich we could in our minds easily derive the various actual forms now living ; but typical also in the newer sense, as that actual form, the limb-skeleton of the latest common ancestors of all Stapedifera, from which have been developed the corresponding parts in all living Stapedifera. The same form belongs to both fore and hind limbs. Using the names applicable to the former, we have, as is well known, humerus, radius and ulna, radiale, intermedium and ulnare, a centrale, and then set around these, five cai'palia followed by their metacarpals and phalanges. Moreover, the strong suspicion of the double nature of the centrale, as evidenced in the descriptions of Cryptobranchns JaponiGus.\ by Schmidt, Goddard, and J. Van der Hoeven, is later, 1865, confirmed by the careful observations and clear presentation of the anatomy of that animal by Hyrtl.J; The Ichthyosaurs§ and * Gegenbaur, Untersuchungen zur vergleichenden Anatomie der Wirbelthiere, Hft. 1, Carpus imd Tarsus. f Gegenbaur, Unters., Hft. 1, p. 57. \ Hyrtl, Schediasma anatomicum. 1865. Gegenbaur, Unters., Hft. 2, p. 165. § Gegenb., Unters., Hft. 2, p. 165, and Jena Zeitschr., Bd. v, Hft. 2, 1870. Gegen- baur, Ueber das Gliedmaassenskelet der Enaliosaurier. In this last a furtlier modifi- cation is made in tlie recognition of the pisiform as the remains of a sixtli row, an(] as being an essential part of the carpus and not merely a sesamoid bone. 294 ./. K. Thacher — Median and Paired Fins. Plesiosaurs give too their unambiguous evidence in the same di- rection. This limb of the latest common ancestors of the Stapedifera, this typical limb of that group, has been named by Huxley, chiropter- ygium* and the term will be found convenient. The (Jhiropterygium and the Fins of Fishes. The homology between the paired fins of fishes and the limbs of Stapedifera has long been recognized ; but the special homologies of the skeletal parts of each has been the subject of much controversy. For a historical sketch of the various divergent opinions endorsed by the highest authorities I must refer to the second volume of Gegen- baur's Untersuchungen. Two pairs of limbs are found throughout the great genetic group of the Gnathostomi. The chiropterygium having been established, the determination of that earlier form typical for all the Gnathostomi became a more pressing question. To an answer to this question the investigations of Gegenbaur now begin to lead. The first part of the second volume of the Untersuchungen dis- cusses the shoulder girdle, and the result is that we are now able to trace clearly and surely the primordial shoulder girdle, the scapulo- coracoid, throughout the Vertebrata. We are no longer in doubt as to what part of the fish fin and girdle corresponds to limb and what to girdle of the Stapedifera. The results of Ge'genbaur's work with which we are hei'e concerned were confirmed by the later but inde- pendent researches of Parker.f Development of the Archipterygimn, Tlieory. This preliminary question having been satisfactorily answered, the derivation of the chiropterygium is next attempted. The second half of the second volume of the Untersiichungen (1865) takes up the pectoral fin of fishes. Quite a number of very excellent figures of numerous Elasmobranchs, Ganoids and Teleosts are given, and the limb skeleton of Protopteras is discussed. The conclusions drawn are as follows : We may take as the most generalized form of limb that of the Elasmobranchs, where its various parts are most plainly presented in * Proc. Zool. Soc. London, IS?*!, pt. i, p. 56. T. H. Huxley, On Ctratodus Forsteri. •j- Parker, Slioulder-girdle and Sternnni. Kay Soc, IMGS. J. K. Tliaeher — Median and Paired. Fins. 295 the rays. It is divided into three parts, Propterygium, Mesoptery- gium and Meta])terygium. Each of tliese consists of a basale, vvliich articulates Avith the shoulder girdle, and a number of rays set on its edge. The fin of Protopteriis is derived from this by the destruction of the pro- and mesopterygium. The metapterygiiim is hei'e represented by the long articulated rod, Avhich alone remains in Liepidosiren paradoxa . The row of cartilages along its sides are the metapterygial rays. In the Ganoids, Polypterns alone has the three divisions repi'e- sented. In this, neither metapterygium nor propterygium bears rays. These are confined to the mesopterygium, which is excluded from the articulation with the shoulder girdle. In the other Ganoids the propterygium fails. Between the mesop- terygium and metapterygium a number of rays are brought into articulation with the shoulder girdle, resembling what is seen in some of the Rays. The Teleosts in the main resemble this second group of Ganoids. The chiropterygium is derived from the metapterygium alone, and thus resembles the limb of Protopterus. The fore limb will serve as an example. The Stammreihe or hasale nietapterygii is presented by the humerus, radius, radiale, carpale radii, the metacarpal and phalanges of the thumb. The other bones are the rays belonging to this, and their arrangement will be best understood by looking at the Ichthyosaurtis limb, fig. 70. In 1870,* Gegenbaur published his explanation of the liml) of the Enaliosaurs. The unbroken lines in fig. 70 of Ichthyosaurus exhibit his view of the relations of the fin with that of fishes. This may be regarded as closing the first stage of the development of the theory in Gegenbaur's publications. The second immediately opens. For in the next numberf of the Jena Zeitschrift there appears an extended article on the ventral fins of Elasmobranchs. The pre\'ious view is modified as follows. The fin-skeleton of the latest common ancestors of all Gnathostomes, is represented pure and simple in the fore limb oi Protopterus a nnectens^ and with only slight modification in the ventral fins of Elasmobranchs. It now has a name given to it. It is called archipterygium. There is a limb gii-dle, complete ventrad. On each side is articulated to this * Jenaische Zeitschr., Bd. v, Hft. .S. Gegenbaur, Ueber das Gliedmaassenskelet der Enaliosaurier, Feb., 1870. f.Ten. Zeitschr., Bd. v, Hft. 4. Gegenbaur, Ueber der Gliedmaassen der Wirbel- thiere im AUgemeinen und der Hintergliedraaassen der Selachier insbesondere, May, 1870. 296 '/ K. Thacher — Median and Paired Fins. the stem-row {Stamm-relhe) , a long taperiug many-jointed cartilaginous rod which bears on the outer side a series of rays. Tliis evidently calls for no change of view regarding the Enaliosaurs or Stapedifera. But the fin-skeleton of iishes exhibits everywhere, except in Pro- topterus and Sct/mnus, a slipping off of the rays from the stem-row" and their articulation with the girdle, and very commonly their artic- ulation with one another and considerable fusion (concrescence). Still another change awaits the primordial limb, even the named archipterygium. In 1871, Gtinther* published his description of Ceratodus. Here the stem-row has a series of rays down each side. The archipterygium is modified to accord with this in the Jena Zeitschrift published x\pril 22, 1872,f where Gegenbaur adopts the " Biseriale Archipterygium'''' as the parent form, and attempts to show that there are some traces of the median row of rays in the pectoral fins of some Elasmobranchs. With the exception of Cerato- dus and the questionable exception of these Elasmobranch pectorals, the biserial has been everywhere reduced to the uniserial form, and still further reduced as heretofore explained. In the third volume of the Untersuchungen, dated May, 1872, a suggestion J: is made of the possible origin of the Archipterygium and the limb-girdles. They are assimilated to tlie branchial arches and their diverging rays, where rays move up upon, and articulate with, the longest middle ray. It is but justice to say that the suggestion is a little vaguely and liesitatingly made. In confirmation of Gcgenbaur's views, Bunge,§ in 1874, published a further investigation of the pectoral fin of Elasmobranchs, showing a number of rays which might be regarded as median, in several species not examined by Gegenbaur. Finally, in 1876, Huxley || took up the question, and, wliile he accepted the archipterygium, he modified the interpretation of a large number of the forms. *Proc. Roy. Soc, 1871, p. 378, and more fully, with a figure of the fin-skeleton, in Ann. and Mag. of Nat. Hist., March, 1871. To these Gegenbaur refers, Jen. Zeitsclir., Bd. vii, Hft. 2, p. 132, note. But a much fuller description is given by Giinther, Phil. Trans., vol. clxi, pt. ii, pp. 511-572. This vs^as pubhshed early in 1872. f Jen. Zeitschr.. Bd. vii, Hft. 2, pp. 131-141. Gegenbaur, Ueber das Archip- terygium. :}; Gegenbaur, Unters., Hft. III. Kopfskelet der Selachier, p. 181, note. 1872. § A. Bunge, Jena. Zeitschr., Bd. 8, Hft. 2, 1874. Ueber die Nachweisbarkeit eines biserialem Archipterygium bei Selachiern und Dipnoern. Bunge also calls attention to the fact that the fringing rays in Protoptems a-i,nectens are on the median (i. e., ven- tral,) side of the axis, and not, as in Elasmobranchs, on the lateral (i. e., dorsal,) side. II T. H. Huxley, Proc. Zool. Soc. Lon. for 1876, PI. 1. On Ceratodus Forsteri. J. K. ThacJier — Median and Paired Fins. 297 Most of the inoditications introduced by Huxley, though perhaps not all, spring from a question which is independent of any theory with regard to the skeleton, archipterygium or other, but which, superior to them, must determine the application of them to the passage fi-om the tisli limb to that of Stapedifera. If an Elasmobranch pectoral fin, for example, of Mustelus, be re- moved and laid on the corresponding hand, with the propterygial edge toward the thumb, and the metapterygial edge toward the little linger, then the ventral surface of the fin will look in tlie same direc- tion as the palmar surface of the hand. But if it be turned over so that the metapterygial edge corresponds to the thumb and the prop- terygial to the little finger, then the dorsal surface of the fin will correspond to the palmar surface of the hand. One or the other of these views must be taken. There is no third possible. Huxley takes the first, Gegenbaur the second. This, how- ever is no new question and no new difference of opinion. Cuvier, following Bakker, named the two ossifications of the scapulo-coracoid which are so generally found in osseous fishes, radius and ulna. Owen simply reversed this nomenclature and Mettenheimer followed him. The question was the same as now respecting the homologies of faces and edges of fin and limb. On the one side, then, we have Bakker, Cuvier and Huxley ; on the other, Owen, Mettenheimer and Gegenbaur. The weight of evidence seems to me to be in favor of the view entertained by the latter group, namely, that the metaptery- gial edge of the fish fin corresponds with the radial or thumb side of the hand, and consequently that the dorsal surface of the fish fin is the palmar (or plantar) surface. But I have no new facts. By reviewing Gegenbaur's work it will be seen that this theory of his rests upon the form of the limbs in the Elasmobranchii and Dipnoi. In the former grouj) it is the hind limbs which furnish nearly all the evidence. The fore limbs (pectorals) are brought in merely to testify to the hiserial character of the archipterygium, of which no Elasmobranch ventral gives a sign. That is to say, the ven- trals having testified to the archipterygium, and that having been accepted, the pectorals find use for themselves in showing thai it was fringed down tlie median as well as the lateral side. If then the same form of limb is found in Elasmobranch and Dipnoan, the same form was undoubtedly possessed by their common ancestors. But as their common ancestors were also undoubtedly common ancestors of all Gnathostomes, therefore all Gnathostome limbs must have been derived from this form. Trans. Conn. Acad., Vol. III. 38 February, 1877. 298 J. K. Thacher — Median and Paired Fins. The testimony of the Ganoids and Teleosts seems to me to be somewhat adverse to the theory. Again it is impossible to think that that of the Stapedifera can be very clearly in its favor, when Huxley, while accepting the archipterygium as the parent form, gives an explanation of the cliiropterygium entirely distinct from and utterly inconsistent with that of Gegenbaur. Any opinion adverse to the archipterygium theory will have diffi- culty in maintaining itself, so long as it does not show that the resemblance between the fins of sharks, and those of Dipnoi is a merely superficial one, and is not able to suggest how a certain show of resemblance might have arisen in two entirely distinct and different series of developments. Another View of the Origin of Vertebrate Limbs. Into competition with this theory, which sees in the fin of Cera- todus that from which all other limbs have been derived, I bring a second which sees in the same only a special development peculiar to the Dipnoi. It is this. As the dorsal and anal fins ^cere specializations of the median folds of Amphioxus^ so the paired fins were specializations of the tioo lateral folds which are supplementary to the median in completing the cir- cuit of the body. These lateral folds, then, are the homologues of the Wolffian ridges, in embryos of higher forms. Here, as in the median fins, there were formed chondroid and finally cartilaginous rods. These became at least twice segmented. The orad ones, wuth more or less concrescence proximally, were prolonged inwards. The cartilages spreading met in the middle line, and a later extension of the carti- lages dorsad completed the limb girdle. If now we seek to determine the form of limb for the Protognathos- tomi, that is to say, for that time for which the archipterygium in its entii'ety is proposed, we should propose this. TTie limbs of the Protognathostomi cimsisted of a series of parallel articulated cartilaginous rays. They may have coalesced somewhat proximally and orad. In the ventral pair they had extended them- selves mesiad until they had nearly or quite met and formed the hip girdle. They had not here extended themselces dorsad. In the pec- toral limb the same state of things prevailed, but was carried a step further, namely, by the dorsal extension of the cartilage constituting the scapidar portion, thus more nearly forming a ring or girdle. This theory naturally diA'ides itself into two parts, namely, the derivation of the Gnathostome limb from a series of sim})le parallel J. K. Thacher — Median and Paired Fins. 299 cartilages, and the derivation of tlie latter from the lateral folds of Amphioxns. Though the last mentioned portion of the theory would derive considerable strength from the establishment of the first, it is not a necessary consequence of it, and the first might be true even if the last were false. If the last be true, of course the first must be true. The establishment of the derivation from the lateral folds of Ainphioxus is made difficult from the al)senee of limbs or anything representing them in the two groups which (in a sense) stand between Amphioxus and the Elasmobranchs, namely, the Myxinoids and Lampreys. As will be seen, it assumes the essential correctness of Huxley's suggestion with regard to the relation between the folds which grow down to inclose the atrial cavity of Amphioxus and the body walls of higher vertebrates. But it is equally consistent with Huxley's* entire suggestion, as put forth by him, or with Ray Lankester'sf modification of it. On the other hand, it is inconsistent with Kowalewsky'sJ view of the homology between these and opercular folds. This must perhaps be consideied still an open question, though Rolph's§ arguments on the other side seem to me of much less weight than they do to Semper. II Yet even if this homology with the lateral folds should have to be giVen up (the embryology of the Marsipobranchs will throw consid- erable light on it), the very frequent occurrence of the formation of external lateral folds parallel to the axis of the body in the bilateral animals in general and in the Vertebrates in particular, renders it quite possible that the paired fins may have had a similar origin. At present, however, I am strongly of the opinion that they are * Joum. of Linn. Soc, vol. xii, No. 59, May, 1875. Huxley, Classification of the Animal Kingdom. f Quarterly Journ. of Micr. Sc, New series, No. 59, July, 1875. Ray Lankester, New Points in the Structure of Ampliioxus. X Mem. St. Petersb. Acad., VII Series, tome xi. No. 4, 1867. A. Kowalewsky, Entwickelungsgeschichte des A mphioxus lanceolatus. § Sitzungsberichte der Naturforschenden Gesellschaft zu Leipzig. .Tahrg. II, No. 1, Jan. 29, 1875. Rolph. Unters. viber den Ban des Amphioxus lanceolatus. See also for a complete account of his investigations, under .,he same title, Morph. Jahb., Bd., ii, Hft. 1, 1876. II C. Semper, Die Verwandtschaftsbeziehungen der gegiiederten Thiere, 1875, p. 317. (Sep. Abdr. aus Semper: Arbeiten a. d. Zoolog-zootom Institut zu Wurzburg, Bd. II). 300 J. K. Thaoher — Median and Paired Fins. derived directly from the lateral folds of Amphioxus. These in their turn may be referred to a reduplication of the process which has already formed the atrial space, but which is not carried so far here, in the lateral folds. Ventral F'm of Aci2')enser. The ventral fin of a young specimen of Acijyenser hreiiirostris is exhibited in PI. LIX, figs. 64 and 05. The fin of one side is separate from that of the other, no synchondrosis uniting the two halves of the girdle. But the part J> approaches closely its fellow. The same separateness of the two sides obtains in the shoulder girdle. The composition of the fin is peifectly evident. Beginning at the aborad end of the row, we have first three separate and parallel rays. The proximal joints increase in length from the first to the last of the three. In the remaining rays these basal joints, increasing still moi'e in length, have united with each other to form the large pelvic piece ab. The composite nature of this is confirmed by the groovings of the surface, which extend about a centimeter before they finally fade out. The iliac process, «, is half a centimeter high. In fig. 64 the proximal joint of the penultimate ray grows up a little under the last ray, in the manner familiar in the median fins. It does not happen to occur in the fin of the other side, fig. 65. The predominant three-fold division obtains. But the penultimate ray in fig. 64 has a tip or an extra segmentation, and c is without the distal segmentation. The breadth and the outline of c raises the suspicion of its double character. I have had >io opportunity of examining other Ganoid fins, and this one of Aci2^enser seems, on the whole, that which most nearly approaches the parent form of the Gnathostomes. But while in the independence of the two sides, in the separateness of the I'ays, and the simple segmentation, it gives us the early form more complete than is elsewhere found, in the number of rays and hi the absence of the iliac process the shark ventrals are less advanced. Elasmohraiich YentraU. We now turn to the derivation of the ventral fins of sharks, one of the two abutments of the Inroad S{)an of the archiptcrygiuru theory. The series of steps by which I conceive them to have been derived J. K. Thaeher — Median and Paired Finn. 301 from the row of ]>arallel rays is presented in woodcuts A, B, C and D. The kind of 'cliange invoked is simply coneresence, with scmie spreading of the cartilage. The former of these processes is abun. dantly shown in the' case of the median tins, while something of the latter process is seen in PI. LI, figs. 12, 18, 14, 15; PI. LIII, fig. 27; PI. LVI, fig. 46 ; PI. LVII, fig. 49. And it is noteworthy that here the rays which jn'olong themselves prox- imally are the orad ones, just as they are in the ventral fins. As for the concrescetice, this has been carried much farther in the dorsal ^n^ oi ^Sq^la- lusAmericanus,3ryliobatis and Raia levis than it has in the shark ventrals. It is barely possible that the definiteness and constancy of the concres- cence in the latter may be in whole or in part determined by the copulatory function of the last part of the fin in male Elasmobranchs. While the derivation of the ventral fins is thus easy from a series of parallel cartilages, we find much greater difficulty in the ease of some of the median fins, in Raia levis, for example, which is, unless my own preconceptions deceive me, a far better case of a biserial archiptery- gium than any furnished in the paired fins, aside from Ceratodics. C Indeed I may state that the origin of this i)aper lay in an observation of a fin of another species of Haia, not however well enough preserved 302 J. K. Thaeher — Median and Paired Pins. for drawing. The very striking similarity to the uniserial archip- terygiuni raised the question whether the median fins, at any rate the dorsal and anal fins, might not have arisen from the same archiptery- ginm. The result of my investigations was a decided negative. It has been absolutely proved that they did not so originate, and the way in which they did originate has been clearly shown. While then, on the one hand it has been shown that the develop- ment of a pair of fins, whoso skeleton consists of a series of parallel rays clothed on each side with a layer of muscle, as a specialization of the lateral folds (raetapleura of Ray Lankester) of AniphioxHs, contains no steps which have not been taken in the same animals in the case of the median fins, so also it has been shown that the development, of the ventral fins and the pelvic girdle of sharks from such a series exhibits no processes or kinds of change which are not also exhibited in the median fins of those same fishes. When we contrast the changes from a series of parallel rays to the completed ventral fin of the shark, as it has been given above, with the changes which Gegenbaur supposes to have made it out of the archiptery- gium, namely, the stripping ofi" of every one of the median rays, for no sign of them is ever found in the ventral fins aside from the Dipnoi, and the slipping off of the orad portion of the rays to immediately articulate with the shoulder girdle, I hardly think that those changes of his will appear so well evidenced as these changes which I believe to have taken place. And when the utter darkness that covers the development of the arehipterygium itself (for it does not seem fair to the arehipterygium to make much account of the suggestion of Gegenbaur respecting the branchial arches, Unters., Hft. iii, p. 181, note) is contrasted with the familiar changes which would have brought these Selachian fins out of the lateral folds Ainphioxus, I hardly think the advantage can lie with the arehipterygium. Homodynamism of Median, and Paired P'ins.* Let us compare the ventral with the dorsal fins, say in Mustelus cams. * Since this paper was written, I have found a paper of Humphrey's on the Homo- logical Relations of Mesial and Lateral Fins of Osseous Fishes, Journ. of A_nat. and Phys., Nov., 1870. Here a comparison between the fins in question is made in the case of the Pike, and the " Iliac " or " Pubic " bones in osseous fishes are assimilated to the interneural spines or to the prominal part of them. Goodsir had made some earlier comparisons without valuable result. See Anatomical Memoirs, vol. ii, p. lOtJ. J. K. Thdcher — Median and Paired Fins. 303 In each there is a hiyer of muscle ou each side of the cai'tilaginous skeleton ; this flat mass is in each divided into separate muscles by septa running from between the skeletal rods straight to the integu- ment, in the way exhibited in PI, LIX, fig. 66 ; in each these muscles develop a tendon in their middle plane parallel to the sides of the fin ; this is inserted in the fascia over the terminal cartilages where the horny fibers begin. These last are the same in nature and arrangement in each. The skeletal elements remain (see PI. LIX, iig. 67). We have a short terminal piece in each, then comes a longer middle piece. There is left in the median fin a proximal row of cartilages, for the most part separate, which are again much shorter than the middle pieces. In the ventral fin the solid basale metapterygii and a half of the girdle correspond to these in every particular, except in not being of separate rays. The similarity between the two fins is complete except in a single particular. And that exception would be removed by a process which is familiar in both, namely concrescence. Even as they stand, I think that a ventral tin with one half the girdle resembles the dorsal more than it does the pectoral. A certain amount of similarity warrants us in inferring an earlier state when the similarity was greater. It has been proved that at that earlier time the median fins were composed of separate rays. The greater similarity, then, can only be attained by the resolution of the basale metapterygii into its component parts. In the Elasmobranchs, as is well known, the primordial fin-skeleton is supplemented in both median and paired fins by the well kno%vn horny fibers. In the higher fishes these are replaced by the dermal rays. The presence of horny fibers in the adipose fin of the salmon, shows that the horny fibers were the earlier form. Therefore the same changes have been taking place in the median and j)aired fins at the same time. The same general result, i. e., of concomitant varia- tion in median and paired fins is confirmed in the sub-groups of Elasmobranchs, with regard to minor changes in the primordial skeleton. The homodynamism of median and paii'ed fins comes out strikingly in Centrlna Salvkmi. Here, according to descriptions, a fold of skin is raised along the median line of the back, recalling the early continuous fold of skin along the back in ^iniphloxus. But similarly there appear two folds of skin along the sides, recalling the continuous lateral folds of Amphioxus. It is at once a proof of the homodynam- ism of the two, and a confirmation of the views here presented of their orisjin. 304 J. K. Thacher — Median and Paired Fins. ■X llie Dlpnoau Fin. In the Teleosts and Ganoids, which are in a sense intermediate between the Dipnoi and Elasrnobranehii, the limb skeleton has been comparatively little modified from the form in this paper set forth as the typical limb skeleton of (Tnatliostomi. JSeurcely any other pro- cesses than reduction of the number of rays and concrescence are con- cerned. Now in the fin of Ceratodns the archipterygium form has certainly been developed, and if tlie previous views be correct, it has been developed from this series of parallel rays. Gilnther* has suggested one way, and Gegenbaurf another, in which a row of parallel rays might transform themselves into an archipterygium form. But it is possible that these fringing rays are new developments. They look very much like it in Protopterus. PetersJ has called attention, in this connection, to the similar struc- ture of the dorsal finlets of Polyptenis, and these might throw some light on the subject. For myself, I am strongly inclined to suspect that the three portions of the second piece of the limb of Ceratodus^ which Gilnther describes, indicate three fin-rays, and that the feather- ing of one of these is a later development. The fact that Huxley could find no sign of division in his specimens seems of little weight in view of the complete fusion which Ave know takes place here and there in median fins. The Limh of Stapedifera. PI. LX, fig. 70 {Ichthyoaauni.s,) exhibits my view of the composi- tion of the limb of air-breathing vertebrates. The dotted lines indi- cate the separate rays of which it is composed. But there are other ways in which it may have been derived from a series of parallel rays, and I oifer this merely as the most probable interpretation so far as I can now see. Fig. 71 does the same for the hind limb of Crypto- branchus Japonicus. The curvature of the rays has been exhibited in a marked degree quite frequently in Elasmobranch median fins. The Innervation of the Paired Fins. I have made complete and definite observations of the innervation only in a single case, namely in the pectoral fin of Mustelus canis. This fin is supplied by the first 15 niyeh»nal nerves together with a very small branch from the vagus. The simplest condition is seen in the aborad nerves. * Phil. Trans., vol. clxi, Pt. ii, p. 534. f Uiiters., Hft. iii, p. 181, note. X MuUer's Archiv, 1845, p. 8. J. K. Thacher — Median and Paired Fins. 305 The aborad four (12-15) branches, coming directly from the myelon, advance, each by itself, close to the metai)teryoiuni, where they each bifurcate, sending one branch to the dorsal and the other to the ventral side of the tin. The next four (S-11) unite to form a rather loose plexus, which separates again into four nerves, wiiich then sub-divide in the same way as the last mentioned four, except that the orad of these behaves a trifle differently in a manner here- after to be described. Now the first seven nerves unite with one another and with the minute branch of the vagus in the following way. The vagal branch emerges from the skull with that nerve, but already rolled up as a separate branch and easily to be separated from it; this joins the first myelonal nerve and this the second, and their sum the third, and so on, until we have a cord formed of the vagal and first seven myelonal branches. This sends ofi' a branch to the muscles and integument in front of the shoulder girdle, but the main part of it proceeds on its way to enter the foramen called by Gegenbaur, liiintriUsoffming^ and then divides within the cartilage of the girdle in the way which he has described, and similarly to the aborad nerves which he has left unnoticed, sending one branch to the dorsal and the other to the ventral muscles of the fin. Now the eighth nerve sends off its ventral branch like those aborad of it, but the dorsal branch enters the entrance-opening with the cord of the vagus and 1-7 spinal nerves; but it does not unite with this cord till after the latter has divided, and then unites with its dorsal branch and emerges with that from the cartilage on the dorsal side of the fin. In another specimen, this dorsal branch of the eighth nerve enters the cartilage by a minute separate foramen, but unites with the dorsal branch of the anterior cord, as in this case. As stated, my observations in the other cases have not been as thorough, and I cannot give the number of ner\es, but in the ventral fin the arrangement is as follows. A number of nerves are gathered together to form the orad cord. This, on coming opposite the fora- men in the pelvic girdle, divides and sends its branch to the ventr^il side of the fin through that. Then the other aborad nerves coming out, each by itself, to the metapterygium divide into two branches for the two sides of the fin, just as in the case of the pectoral fin. This is in 3Iustehis canis. In Eugomphodus Htor(dls, see PI. LX, fig. 00, from the articulation of two or three rays with the girdle, aborad of those which by their concrescence mark themselves out as the pro])terygiuni, it appears Trans. Conn. Acad., Vol. HI. 39 February, 1877. 306 J. K. Thacher — Median avd Paired Fins. that the articulation (or failure of concrescence) of the metapterygium and girdle has taken place farther aborad, and consequently a greater number of rays devote their basal parts to the formation of the pelvic girdle. Expectedly then, we find that the cartilage does in fact spread around the branches of the next two nerves. Through the foramina the ventral branches of these nerves pass, while the branches to the dorsal side pass along to that side above the cartilage. The difference in respect to the relation between the cartilage and the nerves in the pectoral and ventral limb is, that in the former the cartilage thickens so as to include the branching place of the first nerve or bundle of nerves, while in the pelvic limb, it is thinner and merely transmits the ventral branch. In his Memoir on the Shoulder Girdle, Gegenbaur has called attention to the two branching canals or two pairs of openings in the shoulder girdle of the Batoidei. He states that he has not observed whether the aborad one is traversed by a nerve. I have examined this in the case of Raia erinaceus, and found that both fore and after openings transmit nerves in the same way. This is evidently what would be anticipated from what has been herein said. We have here what we had in the ventral fin (and girdle) of Eugoniphodxis ; a greater number of rays are devoted to girdle building, and another bundle of nerves is included in the spreading cartilage. The observations of Rolph on the innervation of Amphioxus^ are in complete agreement with what would be required by the view here advocated. He says " Der ventrale Ast verlauft herab bis in die Seitenfalten. Beim Eintritt in dieselben theilt er sich in zwei Aeste, deren einer {}%^ an der Aussenwand der Seitenfalte hinzieht; der andere durchlaiift die Seitenfalte in querer Richtung, um in die Bauchmuskulatur tiber zu treten, in der er sich nicht weiter verfolgen lassen. Zuvor jedoch gibt er noch einen Zweig ab (^ig), welcher, n^ parallel, an der inneren Wand der Seitenfalte verlauft." The manner of innervation, then, seems to me as totally inconsistent with the Archipterygium theory as it is in thorough and telling har- mony with the view which I have here presented. Addendum. Since the views expressed in the foregoing pages were complete in my own mind six or eight months ago, I had looked for confirmation of them in the brilliant investigations of Balfour on the development *Morph. Jahrb., Bd. ii, Hft. 1, p. 107, 1876. J. K. Tluicher — Median and Paired Fins. ;507 of Elasmobranchs. Tlie preliininary account, however, in the Journal of Microscopical Science, contained nothiuij; bearing on the point, and the pa])ers in tlie Journal of Anatomy and Physiology I have been able to obtain only irregularly. Immediately after the hist proof of the preceding pages had been received, tlie number of that Journal for October, 1876, came into my hands. Here Balfour devotes three or four pages to the limbs. He says : " If the account just given of the development of the limb is an accurate record of what really takes place, it is not possible to deny that some light is thrown by it upon the first origin of the vertebrate limbs. The fact can only bear one interpretation, viz: that the limbs are the remnants of continuous lateral fins.'''' "The development of the limbs is almost identically similar to that of the dorsal fins." He goes on to state that while none of his researches throw any light on the nature of the skeletal parts of the limb, they certainly lend no support to Gegenbaur's view of their derivation from the branchial skeleton. Thus these results have not only been reached independently, but from two different classes of facts. To the belief in the original continuity of the lateral fins and the homodynamism of median and paired fins I was led by observa- tions on adult forms, and particularly on the skeleton. Balfour comes to the same results from embryological investigations, in that group from which on general grounds an answer was most to be expected ; nor do these investigations regard the skeleton. I have also just received the last number of the Morph. Jahrb. It contains a paper by Wiedersheim* confirming Gegenbaur's view respecting the double nature of the centrale. This had previously been shown only in the tarsus of Cryptobranchus Japonicus, (and in the Enaliosaurs). Wiedersheim shows its double character in three Siberian species of Urodela, in both carpus and tarsus. This is a very important confirmation of the chiropterygium, and relieves us of sus- picions with regard to its correctness Avhen we push our inquiries into earlier history and more simple forms. In the same number of the Jahrbuch is a paper by Gegenbaurf on the archipterygium theory. He modifies his explanation of the Stapediferal limb to accord with Huxley's view of the homology of edges and faces of limb and fin. He says that while he does not * Morph. Jahrb., Bd. ii, Hft. 3. R. "Wiedersheim, Die altesten Formen des Carpus und Tarsus der heutigen Amphibien. f C. Gegenbaur, Zur Morphologie der GHedmaassen der Wirbelthiere. 308 J. K. Thachet — Median and Paired Fins. think the correctness of this view fully demonstrated, still he thinks there is a decided balance of probability in its favor. Therefore the ulnar side of the arm now appears as the Stammreihe. In other particulars Gegenbaur reaffirms his previous views. He proceeds to devote considerable space to the discussion of the origin of the archipterygium, and again proposes to assimilate the limb and limb- girdles to the gill-arches with their rays. He supports this sugges- tion with considerable argumentation. To this position the archip- terygium theory leads him. I take this opportunity for expressing my sense of the great advan- tages furnished by the U. S. Fish Commission for the study of marine life on our coasts, and in particiilar by the biological laboratory at Wood's Hole, established in connection with that Commission, and also for acknowledging my personal indebtedness to Professor Baird, through whose courtesy I have enjoyed these facilities for a number of summers. J. K. Tliacher — Median and Paired Fins. 309 EXPLANATION OF PLATES. The figures are all drawn with a camera, and photo-lithographed. They are all three-fourths of the size of the originals, except figs. 1, 2, 3, 4, 20, 21, 22, 60; figs. 20, 21 and 22 are three -eighths, and 60 is a little less than twice natural size. Plate XLIX. Figure l. — Petromyzoa mariiitis. a, Ligamentum longitudinale, or ridge-pole of my- elonal canal ; b, notochord ; c, neural arches. Figure 2. — Petromyzon marinus. a, fin -rays ; &, intermuscular septa. Figure 3. — Petromyzon marinus. a, fin-ray; b, fin-muscles. Figure 4. — Section of Petromyzon marinus, to show the relation of the neural arches to the muscular segments ; a, intermuscular septum ; &, neural arch ; c, blood-vessel ; d, muscular segment ; e, fatty-fibrous ridge-pole of neural canal. Figures 5, 6. — Mustelus canis. First dorsal. Plate L. Figures 7-10. — Mustelus canis. First dorsal. Figure 11. — Mustelus canis. Second dorsal. Plate LI. Figures 12-15. — Mustelus canis. Second dorsal. Figures 16, 17. — Mitstelus canis. Anal. Plate LII. Figures 18, 19. — Mustelus canis. Anal. Figure 20. — Galeocerdo tigrinus. First dorsal. Figure 21. — Galeocerdo tigrinus. Second dorsal. Figure 22. — Galeocerdo tigrinus. Anal. Figure 23. — Eidamia Milberti. First dorsal. Plate LIII. Figure 24. — Eidamia Milberti. First dorsal. Figures 25. 26. — '' " Second dorsal. Figures 27, 28.— " " Anal. Figure 29. — S-phyraa zijgcena. First dorsal. Plate LIV. Figure 30. — Sphyrna zygcena. First dorsal. Figure 31. — '' " Second dorsal. Figure 32.— ' '' Anal. Figures 33-35. — Eugomphodus { Odontaspis) litoralis. First dorsal. Plate LV. Figures 36-39. — Eugomphodus litoralis. First dorsal. Figures 40, 41. — " " Second dorsal. Plate LVT. Figures 42-46. — Eugomphodus litoralis. Second dorsal. Figures 47, 48.— " " Anal. 310 J K. Thacher — Median and Paired Fins. Plate LVII. Figures 49, 50 — Eugomphodus litoralis. Anal. Figures 51-54. — Squalus (Acanthias) Americcmus. First dorsal. Figures 55, 56. — " " " Second dorsal. The spine is re- moved in figs. 53, 55, 57 ; but not in ligs. 51, 52, 54, 56. Plate LVIII. Figure 57. — Squalus Americanus. Second dorsal. Figure 58. — Baia levis. First dorsal. Figure 59. — " " Second dorsal. Figure 60. — Myliohatis Fremenvillei. Sole dorsal. Figure 61. — Acipenser brevirostris. Sole dorsal; a, neural arch; 6, intercalary carti- lages ; c, neural spine ; d, foramen for ventral branch of spinal nerve ; e, foramen for dorsal branch of spinal nerve. Plate LIX. Figure 62. — Acipenser brevirostris. Anal. Opposite 2-| vertebrae. Figure 63. — Section of vertebrarium of Acipenser brevirostris; a, neural spine; 6, ligamentum longitudinale ; c, neural arch. Figure 64. — Acipenser brevirostris, left ventral, from above ; a, iliac process. Figure 65. — Acipenser brevirostris, portion of right ventral, from below. Figure 66. — Section of pectoral fin of Mustelus canis. Figure 67. — Mustelus canis, ventral. Figure 68. — Mustelus canis, pectoral detached from girdle. Plate LX. Figure 69. — Eugomphodus litoralis. Ventrals; a, b and c nerve-foramens. Figure 70. — Pectoral limb of Ichthyosaurus, after Cuvier. Car., carpalia ; Cen., centralia ; u, ulnare ; i, intermediiun ; r, radiale ; U, ulna ; R, radius ; II, humerus. I take, though with some hesitation, Gegenbaur's indentification of the radial and ulnar sides in this limb of IcMhyosaurv,s. Figure 71. — Gryptobranchus Japonicu^s. Hind foot, after Hyrtl, Schediasma anatomicum, 1865. VIII. — The Early Stages of Hippa talpoida, with a note on THE Structure of the Mandibles and Maxillj? in Hippa and Remipes. By Sidney I. Smith. The biological station, established under the auspices of the United States Commissioner of P^ish and Fisheries, at Wood's Hole, Massa- chusetts, during the summer of 1875, afforded several naturalists, and among them the writer, excellent facilities for studying the marine animals of Vineyard Sound and the adjacent waters. The locality is very favorable for obtaining in abundance the free-swimming larvjB of a great variety of marine animals. Among the young of numerous species of crustacea, the zoeae o^ Hippa were particularly interesting, and I svicceeded in obtaining a nearly complete series of the post- embryonal stages of that peculiar genus. Since almost nothing has been published in regard to the habits of any of the species of Hippidaj or Albunidae, a few words in regard to the habits of the only species, of either Fig. i.- of these groups, living upon the coast of New Elngland may" not be out of place here. Hippa talpoida inhabits the entire eastern coast of the United States from Cape Cod southward to the west coast of Florida ; Egmont Key being its most southern and western habitat known to me. At what point it is met or replaced by the Brazilian H emerita, I am unable to determine, never having seen specimens of either species from, or the record of their occurrence in, the West Indies or Central America, although some species of the genus probably inhabits both these regions. On the sandy coasts of the southern United States the H. talpoida is apparently very abundant, while on the coast of New England it is much less common, being found only in special localities, although, * Hippa talpoida, adult female with the antennae extruded, dorsal view, enlarged about two diameters. Trans. Conn. Acad., Vol. III. 40 April, 1877. 3 1 2 S. I. ^%nU7i — Early Star/es of Hippa talpoida. from its gregarious habits, it is usually found in al)undauce in such localities. The northern range of this, as well as of numerous other, southern species is undoubtedly resti'icted by the extreme cold of the winters ; and exceptionally cold seasons probably destroy a large part of the individuals over considerable portions of the coast. Dui'ing the summer of 1870 not a specimen of the adult or half grown Hippa could be found at Fire Island Beach, Long Island, although the extensive sandy beaches of that region offer specially favorable localities, which were thoroughly searched ; but during the last of August and early September, the young just changed from the zoea, and also in a little later stage, appeared abundantly upon the beaches. During the following summer no fully grown specimens were found on the shores of Vineyard Sound, though half grown specimens (perhaps from the young of the previous season) were common. During the summer of 1875, fully grown specimens of both sexes were found in great abundance at a single, very restricted locality near Nobska Point, on the shore of Vineyard Sound, although at this time none could be foimd at the particular locality where they were common in 1871. Upon our shores, as far as I have observed, the Hippa inhabits sandy beaches which are somewhat exposed to the action of the waves. It seems to prefer only a narrow zone of the shore, at or very near low water mark, where it lives gregariously, burrowing in the loose and changing sands. At the locality near Nobska Point above referred to, it was obtained in great abundance by digging over the sand just at the edge of the receding waves. Several individuals were often thrown out at a single stroke of the spade, but the won- derful rapidity with which these animals burrow made it extremely difficult to secure more than one or two of them at a time. The smooth, oval form of the animal, with the peculiar structure of the short and stout second, third, and fourth pairs of thoracic legs, enables them to burrow with far greater rapidity than any other crustacean 1 have observed. Like many other sand-dwelling crustaceans, they burrow only backwards ; and the wedge-shaped posterior extremity of tlie animal, formed by the abrupt bend in the abdomen, adapts them admirably for movement in this direction. When thrown upon the wet beach, they push themselves backward with the burrowing thoracic legs and, by digging with the appendages of the sixth segment of the abdomen slightly into the sui-face, direct the posterior extremity of the body downward into the sand. Upon the beaches, at least where there are any waves, they seem usually to be buried completely S. 1. S)i}!f/i—H<(rh/ Stdf/ex of Ilippa talpokht. :UM beneath tlio surluce. Occasionally, however, they are found swinnnin^- about in ])ools left by the tide, and they undoubtedly, when undis- turbed, sometimes come out and swim in the same way along the shore, though they probably never venture far from tlie bottom. When first placed in an aquarium with a few inches of sand at the bottom, they invariably ])lunged at once entirely beneath the sand, but, after a few moments of quiet, usually worked themselves gradually towards the surface, resting in a nearly perpendicular position with just the tips of the aiitennuhe and eyes at the surface, while the ex- cui'rent water from the branchiae formed a small opening and a slightly boiling motion in the sand. Occasionally, when entirely undisturbed, they would suddenly leave the sand and swdni rapidly round the top of the aquarium for a moment and then dive suddenly to the bottom and bury themselves in the sand. In swimming, as well as in burrowing, the telson was carried appressed to the sternum and they invariably moved backward, the motion being ap|)arently produced by the appendages of the sixth abdominal segment and the anterior thoracic legs, while the latter served also as steering organs. During all the ordinary motions of swimming and burrowing, I have never seen the antenna^ extruded, although the peculiar arrange- ment of the peduncular segments and their complex system of muscles are apparently specially adapted for extending and withdrawing these beautiful organs. When the animals are thrown into alcohol however, the antennre are sometimes thrown out convulsively and then immediately retracted. In life the antennae are most of the time held in the position in which they are usually found in alcoholic specimens, that is, between the second and external maxillipeds, with the peduncles crossed in front, and the flagella curved down and entirely round the mouth so that their dense armament of sette all project inward. When extruded, the distal segments of the peduncle are revolved half way round on the proximal ones, so as to carry the whole appendage to its own side of the animal and throw the curve of the flagellum into a reversed position. Judging from the pecul- iarly armed setae, of the flagella, one of the principal offices of the antenna? is the removal of parasitic growths and all other foreign substances from the appendages of the anterior portion of the animal. The mouth parts of the adult are not adapted for ordinary prehen- sion or mastication, but I am unable to make any positive statement in regard to the food of these animals. In all specimens examined the alimentary canal was filled with tine sand which seemed to be nearly free irom animal or vegetable matter. The material from the 314 ^S', /. tSnuth — EarJy St<i<j(>.s of Illppa Udpoidn. stomaeli, however, shew, under the microscope, u small quantity of vegetable matter, and it seems probable that the sand is swallowed for the nutritive matter it may contain. Upon the beaches of Vineyard Sound the two sexes appeared to occur in about equal numbers, although in museum collections the males are often rare. This is probably due to the great inequality in size between the male and female, the length of the carapax in the larger females from Vineyard Sound being 20 to 22""", while in the largest males it does not exceed 14"'"'. The sexes differ also in the form of the telson (Plate XLVIII, figs. 7, 8) which is narrower and more triangular in the male than in the female. Females carrying eggs were found during the entire month of August, and during that period the embryos within the eggs were nearly fully developed in many of them. Undoubtedly, however, the term of carrying eggs extends over a much longer period than this. The eggs are nearly spherical, "40 to -45"'"' in diameter, and the yolk mass is orange yellow while the formed tissues of the embryo are nearly colorless. Numerous attempts to obtain newly hatched young, by keeping egg-carrying females in aquaria, failed from the parent invariably casting off the eggs before they were fully matured. Consequently I failed to secure the earliest stage of the zoea, for the youngest individuals taken in the towing net were apparently in the second stage. Very nearly fully developed embryos, when removed from the egg, were found to possess all the normal articulated appendages of the fully formed zoeae, but there was no appearance of lateral spines upon the carapax and the rostrum was broad and obtuse. In this stage the embryo agrees almost perfectly with the figure of the zoea of Ilippa emerita from the coast of Brazil, given by Fritz Mliller in his work entitled " Ftir Darwin."* The difference between the embryo in this stage and the second zoea-stage (Plate XLV, fig. 1), in which * English translation, London, 1869, p. 54, fig. 25. The figure is accompanied by the following paragraph : " The Zoea of the Tatuira [Hvppa\ also appears to differ but little from those of the true Crabs, which it likewise resembles in its mode of locomo- tion. The carapax possesses only a short, broad frontal process ; the posterior margin of the tail is edged with numerous short setse." This, as far as I am aware, is the only published account of the development of any sjDecies of Hippidse, except a note by myself (in an article on " The Metamorphoses of the Lobster and other Crustacea," in the Report of the LT. S. Commissioner of Fish and Fisheries, Part J, 187."5, p. 530) recording the occurrence, at the surface in Vineyard Sound, of the young in what is described further on in these pages as the megalops-stage. S. r. SntlfJi — KitrJj/ Stat/es <}f Tfijypa t<dpoi(la. 315 tlio rostrum and lateral spines are enormously developed, suggests the possibility that MtlUer had observed only imperfectly developed young zoea^ in which the rostrum and lateral spines were not ex- pamled. It seems scai'cely probable that such a difterence could exist between the first stage of the zoea, when the veiling membrane, in which, on first escaping from the egg, the young are usually enveloped, has been entirely cast oft' and the lateral spines and the rostrum are fully expanded, and tlie second zoea-stage about to be described. The three later, true zoea-stages obtained are evidently contiguous steps in the development and are here designated the second, third, and last stages of the zoea. From this last stage the zoea passes at once into a stage closely resembling the adult in gen- eral form, but with the eyes still very large and the abdomen furnished, with powerful swimming legs. This condition of the animal corres- j)onds perfectly to the Brachyuran megalops and may properly be designated as the megalops-stage. Second zoea-stage. In this stage the young (Plate XLV, fig. 1, ventral view) are a little over 8""" in length, from tip of rostrum to the posterior margin of the carapax, and a little over 2"'"' between the tips of the lateral spines. In general form the carapax is oval, with the smaller end for- ward, and its surface is very smooth and regularly rounded. The dorsal surface of the carapax is strongly convex but very regularly rounded and wholly devoid of any rudiment of a dorsal spine, which is so generally characteristic of the zoeae of Brachyura. At the bases of the ocular peduncles the carapax is shai-ply contracted laterally into an exceedingly long^ very slender, and slightly tapering rostrum curved regularly downward until, toward the tip, it becomes nearly parallel M'ith the posterior margin of the carapax. The lateral spines are nearly as long as the diameter of the carapax, are situated far back and low down on the sides of the carapax, and are directed downward and obliquely outward, but are not strongly curved. Beneath, the carapax ciirves inward on all sides, leaving a compara- tively small opening which is wholly inferior, with its anterior j)ortion about as broad as the telson, but posteriorly contracted into a narrow abdominal sinus, of which the rounded posterior margin is nearly on a line between the lateral spines. This shortening of the inferior opening, carrying the abdomen forward and wholly beneath the carapax, together with the absence of the dorsal spine, gives the animal an apj^earance unlike ordinary Brachyuran zoese. 310 iS. I. /Smith — Early Stages of Hippo talpoida. The ocular peduncles are stout, regularly tapering to near the bases, and are usually carried perpendicular to the mesial plain, though they admit of considerable motion in all directions. The cornea is considerably larger than the diameter of the peduncle, its diameter being nearly a third of the horizontal diameter of the carapax, and, when the peduncle is held straight out, reaches slightly beyond the lateral margin of the carapax. The antennulae (Plate XL VI, tig. 1) are still rudimentary, simple, sackdike, unarticulated appendages, tapering towai-d the tip, which is furnished, as usual in this stage of development, with three stout, filiform, obtuse setne, diftering slightly in length, diameter, and amount of curvature, and of which the longest is about half the length of tlie antennula itself. The antennse (Plate XLVI, tig. 2) are of about the same length as the antennula-, but of nearly the same diameter throughout, and are armed distally, at the outer edge, with an acute, dentiform process (a, fig. 2) directed straight forward and itself armed with a minute, setiform spine on the inner edge near the tip. Between the base of this process and a slight, rounded prominence (c, fig. 2), situated at the extremity of the inner margin, and which represents the rudiment- ary flagellum, there is a similar, but slightly more slender, process (b) attached at its base by an oblique articulation and armed, near the tip, with a minute, setiform spine like that upon the outer process. The oral appendages differ very little from their condition in the last zoea-stage, under which they are fully described. The labrum and labium differ scarcely at all, except in size, in the three zoea- stages here described. The labrum, as seen from beneath, is a broad, somewhat triangular prominence between the bases of the antennulae and the tips of the mandibles. The labium is deeply bilobed, though far less deeply than in the adult, with the lobes broadly rounded and the entire margin clothed with microscopic hairs. The mandibles are nearly as in the last zoea-stage. They are stout at the bases, but taper to very slender tips, which are only slightly different on the right and left side. There is no molar area, but the crown of the mandible is longest in a vertical direction and is armed inferiorly with four long, but blunt, teeth which decrease rapidly in size as they approach the middle of the crown, where they are met by a series of six or seven long, slender, setadike processes which occupy the superior half of the coronal margin. The first pair of maxilhie (Plate XLVI, fig. 11) are symmetrical and composed of the same parts as in the adult. The inner lobe S. T. Snrit}) — J^arly Sfafies of Hippo, talpotda. '.\\1 {a, fig. 1 1) is small and ti|)])t'(l with tliroc long seta\ Tlie outer lobe [h, fig. 11) is broader than tlie inner and armed at the extremity with three nearly equal, long and slender teeth, of which the distal one appears like a process from the margin, showing no line of artic- ulation at its base. The palpus ('*, fig. 11) is very small, composed of a single segment and tipped with a long, plumose seta. The second pair of maxilla? (Plate XLYII, fig. l) are very imperfect. The protognath (</, fig. 1) is a small, obtuse lobe tipped with three short setse. The scaphognath (<■% fig. 1) projects beyond it anteriorly as a slightly larger lol^e, while posteriorly it is broad but short and truncated, and the anterior lobe and the oi;ter edge are, as yet, alone furnished with seta?. The first and second pairs of maxillipeds, or natatory legs, (Plate XLV, fig. 1, second pair) are similar in structure to those of most zoea^ and differ only slightly from each other. In both pairs the basal portion, or protognath, is alike stout, about as long as the exognath, and unarmed, except by three or four minute setfe on the distal portion of the inner margin. The exognaths, or natatory branches, alike in both pairs, are nearly cylindrical, but flattened at the tips, where they each bear a series of eight slender, plumose setae, which, in the middle, are as long as the exognath itself but decrease in length to the outer ones, which are scarcely more than tw^o-thirds as long. The inner branch, or endognath, in both pairs, is composed of four cylindrical segments subequal in length. In the first pair, however, the inner branch is shorter than the exognath, while in the second pair it is considerably longer. In both pairs the three proximal segments of the endognath are each ai-med with two or three small setje on the inner side, and the distal segment, which is much more slender than the others, is tipped with four setae, of which two are nearly as long as the segment itself and pectenated with minute, setiform spinules along one side, and the two others shorter and apparently unarmed. The third pair of maxillipeds and the four antei-ior pairs of thoi-acic legs are, even at this early state, represented by a series of clearly defined, though entirely nnsegmented, processes situated just above and back of the bases of the second maxillipeds and entirely wnthin the carapax, but visible through it, in a lateral view of the animal, in a line nearly parallel with the posterior margin of the carapax. In the single specimen examined, no lobes representing the slender, posterior thoracic legs of the adult could be discovered. Above each of the processes representing the first four pairs of thoracic legs there is a minute, papilliform process apparently representing one of the branchial appendages belonging to these legs in the later stages. 318 S. I. Smith — hJiirlj/ SUiffcH of Ilippa tdlpoiihi. The iilxlouu'i) is smaller aiul has much less freedom of motion in the mesial phmc than in most Brac^hynran zoea^ The first segment is not clearly (litt'erentiated from the thorax. The second, third, fourth, and fiftli segments are entirely without ap))endages; the second and third are nearly equal in length and sub-cyrnuli-ical ; the fourth is slightly shoi'ter and is exj)anded considerably at the posterior ex- tremity ; and the fifth is about as long as the fourth, compressed vertically, and broadly exj)anded at the posterolateral angles so that it is about twice as broad as long. The sixth segment is consolidated with the tclsoii, forming a broad, lamelliform, caudal a])pendage about as long as tlu' middle breadth of the caraj)ax. The a])pcndages of the sixth segment (Plate XLVIII, fig. 10) ai-e small, rudimentary, Oppressed to the under side of the telson so as to be hidden from above, and are eac^h (composed of a stout basal segment and a single narrow lamella (the outer) tipped with two slender setse, of which the outer is about as long as the lamella itself and the inner much longer. The lateral nuirgins of the telson are slightly curved outward aiul unarmed, but project )»osteriorly into a stout tooth each side of the strongly arcuate ]»osterior margin. This postei ior margin has, in all tile zoea-stages here described, a remarkably complex armament of ciliated spines and miuute teeth (Plate XLVIII, tigs. 1:5, 14, 15). In a considerable iiumbei- of specimens in tlu' third and the last stages, the mnnber of these ciliated sj)ines is usually twenty-six, of whicli the eighth, counting (Voni either side, is the largest, and the sixth and tenth usually the next in size. One of the specinu'us in the second stage (Plate XLVIII, iig. 1:5) conforms stiiiily with this: there arc two sub-nu'dian spines (r/, </, Iig. i;5) separated by a single denticle, then each side a slightly larger sj»ine (A, h) separated from the sub-nu'dian ones by a single denticle, then four alternately smaller and l:irg(>r spines [c, d, ('.,/') se|»arated from each other by two denticles at each intors])ace, the outer (/') of these four sjtines beino" the eighth, c(»unting from either side, ami the largest. Outside this large sjtine there arc, each side, seven smaller sj)iiu's separated by interspaces which increase toward the outer margin and are armed with from two to nine denticles. The space between the outer spine each side and the tooth of tlu' lateral margin is gri'ivtei- than any of the interspaces bctwi-en the spines and is armed with twelve or thir- teen denticles. In the other specimen in this stage there are only twenty-five spines, a single niediiin spine (Plate XIA'III, tig. 14,^^) taking the place of the two sub-nu'dian spines and the denticle between them ; otherwise the Sj)ini'S and denticles are essentially as in S. I. S/tr/'f/i — Earl;/ Stages of Hippa talpolda. 310 the first specimen and as in tlio succeeding stages. This is appar- ently an abnormal variation in the armament of the telson. The arrangement of these spines, and especially whether they be odd or even in number, I have usually found a constant character for distin- guishing the larval forms in different groups of Podophthalmia. Of this stage only two specimens Avere secured, one taken on the evening of September 4, the other in the day-time the next day. In coloration and habits they agreed essentially with the young in the succeeding zoea-stages. Third zoea-stage. In general form and a]»]K'arance the zoeas in this stage very closely resemble those in the second, although they have increased considerably in size, and especially in the length of the rostrum, which is relatively longer and more slender. They are about 4"5""" in length, from the ti]) of the rostrum to the posterior mai'gin of the cara})ax, and nearly :V""' between the tips of the lateral spines. The ocular peduncles and eyes have increased only slightly in abso- lute size and are relatively smaller than in the second stage. The antennuhe liave changed very little. There is, as yet, appar- ently no distinction of peduncle and fiagellum, although the two or three distal segments of the latter are faintly indicated, and, on the inner side, there are two or three filiform setae on tlie penultimate segment in addition to the three on the terminal segment. The antennae (Plate XLVI, fig. ;3) have increased in size but show no indication of segmentation. The two dentiform processes (a, ft, fig. 3) have each two or three minute spir.ules at the tip, but are otherwise unchanged ; the fiagelluin (c, fig. 3), however, has increased so as to project beyond the tips of the dentiform processes and show ))laiiily its true character. The labrum, labium, mandibles, and first maxillse, excej)t in size, do not differ appreciably from their condition in the second stage. The second maxillae differ but little, the scaphognath being a little more elongated posteriorly, so as to approach slightly its form in the next stage. The first and second pairs of maxillipeds differ from those of the first stage only in the exognaths, which are each furnished with ten instead of eight terminal seta'. The lobes representing the third pair of maxillipeds and the four anterior pairs of thoracic legs have increased much in size, are Trans. Conn. Acad., Vol. III. 41 April, 1877. 820 S. I. Smith — Early Stages of Hippa talpoida. curled up closely beneath the sternum, and, in all the specimens examined, show the segments of the succeeding stage faintly indi- cated within. The posterior thoracic legs appear to be represented by a small process each side, just back of, and nearly hidden by, the rudimentary fourth pair. Four pairs of gills on each side are repre- sented by two slender processes, one above the other, at the bases of each of the four anterior pairs of legs The proximal segments of the abdomen are almost exactly as in the previous stage, except there are very slight elevations beneath the second, third, fourth, and fifth segments, where the rudimentary legs are to appear in the succeeding stage. The sixth segment is still consolidated Avith the telson. Its appendages (Plate XLVIII, fig. 11) have increased much in size and the inner lamella (c, fig. 11) has appeared as a small, sack-like appendage at the base of the outer lamella {b, fig. 11), which is twice as long as in the previous stage, very narrow, only slightly expanded in the middle, and sub-truncate at the extremity, where it is furnished with four slender and curved setoe, of which the median ones are longer than the lamella itself, while the outer are little moj-e than half as long. The telson is of the same form as in the previous stage and has the same number of ciliated spines in the armament of the posterior border, while the number of denticles in the interspaces has considerably increased, though they are not as numerous as in the succeeding stage. In one specimen there is the same abnoi-mal arrangement of the spines described under the second stage, that is, there are only twenty-five spines in all, one median spine taking the place of the two sub-median spines and the denticles separating them. The young in this stage were taken on several occasions, both in the day-time and evening, from August 28 to September 8. Their habits and coloration in life were the same as in the last zoea-stage. Last zoea-stage. The length from the tip of the rostrum to the posterior margin of the carapax and the breadth between the tips of the lateral spines are nearly twice as great as in the second stage, while the rostrum is relatively considerably longer than in either the second or the third stage, its entire length being nearly twice that of the carapax proper. The general form and appearance of the young in this stage are shown upon Plate XLV, figs. 2, 3, 4. The eyes and ocular pediincles are very little larger absolutely than in the last stage. The diameter of the cornea is scarcely a S. I. Smith — E((rh/ Stagefi of Hippa talpoida. :i21 fourth the horizontal diameter of the carapax and the ocuhir pedun- cles are slightly shorter, proportionally, than in the earlier stages. In the antennulffi (Plate XL VI, fig. 5) the segmentation of the tlagellum is carried nearly or quite to the peduncle, which, however, shows no division into segments and no clear separation from the flagellum. The ilagellum itself is composed of six or seven segments which are a little broader than long and of which the terminal one is furnished with three filiform setae, the penultimate and antepenulti- mate with two or three each which are situated upon the inner side at the distal articulations, while there are two similarly situated, hut small and i-udimentary setoe, upon the fourth segment from the tip. The antenna? (Plate XLVI, fig. 4) have increased very much in size, and the flagellum is much longer than the peduncle. The peduncle shows but one distinct articulation, which is near the bases of the dentiform processes and apparently represents the articulation between the second and third segments of the fully developed appendage. The dentiform processes («, &, fig. 4) ar-i much more slender and propoitionally smaller than in the third stage, but are armed with the same number of spines at the tips. The flagellum (c, fig. 4) externally shows no indication of segmentation, but, in all the specimens examined, the articulations of the flagellum of the succeeding megalops-stage is distinctly visible beneath the integu- ment, as shown in the figure. The labrum (Plate XLVI, fig. 5, J), as seen from beneath, is a conspicuous, somewhat ti'iangular prominence between the bases of the antennula? and the mandibles, with the margins and the inferior surface regularly rounded and without emarginations at any point. The labium (t?, fig. 5) is deeply bilobed, the regularly rounded lobes projecting each side of the oral opening nearly to the tips of the mandibles and having the edges clothed with microscopic